Many sources indicate that the European populace of the early middle ages practiced poor hygiene. This societal norm may have influenced the development of Minhag Ashkenaz as well as the halachic views of Tosfot. Three such instances will be suggested.
Christian vs. Muslim Hygiene Standards
Unlike the religion of Islam  and Muslim physicians  that stressed the importance of personal hygiene, many early Christian leaders placed emphasis only on the cleanliness of the soul and not the body:
Many early saints embraced filth enthusiastically and ingeniously. St. Agnes (Rome, c. 291 – c. 304) never washed any part of her body during her admittedly short life of thirteen years. Godric, an English saint (1065 – 1170), walked from England to Jerusalem without washing or changing his clothes. … At home in his hermitage in the woods near Durham, he wore a hair shirt which, when combined with summertime sweat, supported abundant lice. St. Francis of Assisi (Italian Roman Catholic friar, 1182 – 1226) revered dirt and was said to have appeared after his death to compliment friars on their grubby cells. 
Ulrich, an abbot of Cluny, France and Regensburg, Germany (1029 – 1093) said: “As to our baths, … there is not much that we can say, for we only bath twice a year, before Christmas and before Easter.” 
The early medieval hygiene we know most about was that practised by monks, who were not only literate but eager to document the monastic life … Other orders allowed three baths a year, before the feasts of Christmas, Easter and Pentecost, but monks whose holiness trumped cleanliness could decline any or all baths. Three a year represented a level of cleanliness that was probably below the upper-class standard of the day but above that of the peasantry. 
Further evidence is from Thomas Aquinas (Italian Dominican friar, 1225 – 1274):
So spiritual a character as St. Thomas Aquinas approved of incense in church
because it masked the prevailing body odour, which, he admitted, “can provoke
Secreta Secretorum, a 10th-century Arabic treatise on science and medicine translated into Latin, was one of the most widely read texts of the 11th through 13th centuries among European intellectuals . It included a section on bathing recommendations, about the seasons in which they should or should not be taken. The author writes that bathing in the summer should be avoided as much as possible .
As medieval times progressed hot baths did reach Western Europe. However, Jews were only allowed to visit them once a week:
The hot air and vapor baths of the Byzantine peoples were adopted by the Mohammedans, and later on made known to the peoples of Western Europe through the Spanish Arabs and the Crusaders. … Their statutes are well known to us. The Jews were allowed to visit them once a week. [8a]
The people of Spain, however, were generally cleaner than the rest of the Europeans:
… the cleanest corner of early medieval Europe was Arab Spain. Unlike in Christianity, cleanliness was an important religious requirement for the Muslim, and a ninth-century observer described the Andalusian Arabs as “the cleanest people on earth.” While the Christians in the north of Spain “wash neither their bodies nor their clothes which they only remove when they fall into pieces,” a poor man in the Arab south would reportedly spend his last coin on soap rather than on food. Arab Spain sparkled with water—in pools, fountains and hamams. Every neighbourhood had its public bath. When the Christians recaptured Cordoba in 1236, the city had three hundred hamams as well as hot and cold water in private bathrooms. 
In the 10th century, antiseptics were being used in hospitals in Islamic Spain. Special protocols were used to keep hygiene before and after surgery. 
(1) Thirty Days of No Bathing for the Mourner
The Talmud only forbade bathing for a mourner for seven days:
תעניות יג: הלכתא אבל אסור לרחוץ כל גופו בין בחמין בין בצונן כל שבעה
The Talmud’s ruling is followed by Maimonides:
רמב”ם הלכות אבל פרק א הלכה א
מצות עשה להתאבל על הקרובים, שנאמר ואכלתי חטאת היום הייטב בעיני ה’, ואין אבילות מן התורה אלא ביום ראשון בלבד שהוא יום המיתה ויום הקבורה, אבל שאר השבעה ימים אינו דין תורה,
פרק ה הלכה א
אלו דברים שהאבל אסור בהן ביום ראשון מן התורה ובשאר ימים מדבריהם: א אסור לספר, ולכבס, ולרחוץ…
The Shulchan Aruch similarly followed the Talmud and Maimonides:
שולחן ערוך יורה דעה הלכות אבילות סימן שפא :
רחיצה כיצד, אסור לרחוץ כל גופו, אפילו בצונן;
The Rema however, cited the custom and ruling of early Tosfot:
וכל זה מדינא אינו אסור רק שבעה, אבל אח”כ מותר ברחיצה, אלא שנהגו האידנא לאסור (כל) רחיצה כל ל’ יום (מהר”מ הלכות שמחות ובהגמי”י פ”ז מהל’ יו”ט ופ”י דהל’ אבל); ואפילו לחוף הראש אסור (לקמן סי’ ש”צ ובא”ז); ואין לשנות המנהג, כי מנהג קדום הוא ונתייסד על פי ותיקין (א”ז בשם רשב”א).
Darkei Moshe quoted R’ Yitzchak Or Zarua (Bohemia, 1200 – 1270) as the first source where this custom is recorded:
וזה לשון אור זרוע … סוף הלכות אבלות בתשובה (סי’ תנא) דנהגו לאסור לרחוץ כל שלשים משום גזירת כיבוס ואסור לשנות המנהג עכ”ל
Perhaps the true origin of this custom was the infrequent bathing practices common in early medieval Europe. As it was usual for people to not bathe for a week’s time, the Talmud’s one-week bathing restriction was not meaningful for a mourner and therefore the restriction was extended to a full 30 days.
(2) The Reason Why Hot Full-body Bathing on YomTov is Forbidden
Any activity prohibited on Shabbat is permitted on Yom Tov if that work is necessary for daily sustenance – “ochel nefesh”. The Talmud (Ketubot 7a) establishes a principle that any type of activity that is applicable to ochel nefesh is permitted even when that activity is not being used for the sake of food, provided that it is shaveh lechol nefesh – of common pleasure enjoyed by all or most people.
In the Mishna (Beitzah 2:5), Bet Shammai say that one may only heat up water on YomTov for washing one’s feet, and only so to a temperature that one would be able to drink; Bet Hillel permit heating up the water for one’s feet even as hot as needed to bathe:
בית שמאי אומרים לא יחם אדם חמין לרגליו אלא אם כן ראויין לשתיה ובית הלל מתירין
Tosfot (Beitzah 21b s.v. Lo Yicham) followed by most European Ashkenazic authorities [10a], wrote that Bet Hillel only allowed heating water for one’s hands and feet because that is a pleasure enjoyed by everyone – shaveh lechol nefesh. However, in Tosfot’s view, bathing one’s entire body Bet Hillel did not regard as shaveh lechol nefesh as only pampered individuals required such extensive full cleansing – accordingly heating water for a body bath on YomTov was prohibited according to Torah law:
(ביצה כא: תוספות ד”ה לא)
דדוקא לרגליו אבל לכל גופו אסור דדבר השוה לכל נפש בעינן וזה אינו ראוי אלא לבני אדם מעונגים אבל ידיו ורגליו שוה לכל נפש
Tosfot may have interpreted the words of Bet Hillel through the lens of early northern European Middle Age hygienic standards. In their world, daily full-body bathing was not shaveh lechol nefesh – a common need for most people. Tosfot, therefore, assumed that Bet Hillel forbade full-body bathing on YomTov because it was not shaveh lechol nefesh.
Rabbis in countries under Muslim dominion or influence were accustomed to much higher cleanliness standards. To the minds of the Iraqi Gaonim (cited in MB Shaar HaTziyun 511:8) and medieval Spanish authorities, the daily body shower was certainly shaveh lechol nefesh and Bet Hillel had an alternate reason for prohibiting hot full-body bathing on YomTov.
Rambam (b. Cordova, Almoravid Empire, Spain 1135, d. Egypt 1204) wrote that Bet Hillel’s bathing restriction was merely a rabbinic prohibition – lest such bathing lead one to bathe in a bathhouse (which involves the transgression of many melachot on Shabbat and several melachot on YomTov). (Laws of YomTov 1:16):
לפיכך מחמין חמין ביום טוב ורוחץ בהן פניו ידיו ורגליו, אבל כל גופו אסור משום גזירת מרחץ
According to Ritva (b. Seville, Spain, 1260s – 1320s) daily bathing was shaveh lechol nefesh and was only forbidden by Bet Hillel for another reason:
ומיהו להחם חמין לכל גופו אסור אף על פי שהוא שוה לכל נפש, דגזרו בו משום מרחץ שיש כמה מלאכות אסורות כדאיתא בשבת פרק כירה (מ’ א’), כן פירש הריטב”א ז”ל
Ramban (Girona, Catalonia, Spain, 1194–1270), in his commentary to Shabbat 39a, wrote disapprovingly of the French Tosfot’s underestimation of the necessity of daily bathing:
ויש כאן שאלה כיון שרחיצת כל הגוף וזיעה כולן מותרין ואפי’ להחם בי”ט, למה גזרו כלל עליהן בי”ט ומאי איסורא אתי מינה. ואם תאמר משום שבת, אין ראוי לגזור י”ט אטו שבת בדבר של אוכל נפש, וי”ל כיון דבי”ט נמי איכא בהו איסור כגון מדיח קרקע וסך א”נ סחיטה דאלונתית וכיוצא בהן עשו י”ט כשבת לכך ואינו מחוור, ובתוספות מפרשים רחיצת כל הגוף אינה צורך כל נפש ומן התורה הוא אסור … ואלו דברי נביאות הן שרחיצת כל הגוף תאסר… ואדרבה הנאת כל הגוף טפי צריכא וטפי שריא…
The Mishna Shabbat 3:4 describes the Jewish inhabitants of Tiberias who laid a pipe of cold water through their hot springs to enable themselves to take hot baths on Shabbat and YomTov. This story demonstrates that the Tiberians of the general era of Bet Hillel were accustomed to daily bathing and did not want to go through even one day without their bath. They surely regarded such activity as shaveh lechol nefesh!
(3) The Custom of Not Bathing During the Nine Days
According to the Talmud (Taanith 30a) even on the eve of Tish’a B’Av one can bathe in hot water:
ר׳ ישמעאל בר׳ יוסי אומר משום אביו כל שעה שמותר לאכול מותר לרחוץ
Maggid Mishna (on Maimonides’s Laws of Fasts 5:6):
ודבר ברור הוא שבדין גמור אפי’ ערב תשעה באב מותרין לרחוץ:
R’ Yaakov Ba’al ha-Turim (b. Cologne, 1269 – d. Toledo, 1343) in his Tur (OC 551) citing Ra’avyah (Germany, 1140–1225) recorded a nine-day non-bathing custom as the Minhag Ashkenaz:
כתב אבי העזרי נהגו אבותינו שלא לרחוץ מר”ח ויש עלינו לקיים משום אל תטוש תורת אמך
Rambam (Laws of Fasts 5:6) wrote of a shorter custom amongst Sephardic Jewry – to refrain from bathing only during the week of the 9th of Av until after the fast. This abstinence period varies each year but is at least four days shorter than the Ashkenazic full nine-day custom:
משיכנס אב ממעטין בשמחה, ושבת שחל תשעה באב להיות בתוכה אסור לספר ולכבס וללבוש כלי מגוהץ אפילו כלי פשתן עד שיעבור התענית, ואפילו לכבס ולהניח לאחר התענית אסור, וכבר נהגו ישראל שלא לאכול בשר בשבת זו ולא יכנסו למרחץ עד שיעבור התענית
It may be that the Ashkenazim believed extending the non-bathing period to a full nine days was appropriate because many Europeans regularly refrained from bathing for days at a time anyway. This is better understood together with the recommendation of the popular Secreta Secretorum (cited above) to avoid bathing in the summer. An extension of the bare Talmudic requirements was necessary to make any mourning perceptible.
Rav David Bar-Hayim argued similarly in his “A Brief Guide to the Three Weeks and the Nine Days” (pg. 2 – 3, 30 July 2014 http://machonshilo.org-3weeks):
Question: What of the minhagh of not bathing during the Nine Days?
Answer: Hazal did not forbid bathing during this period, with the exception of Tish’a b’Av itself. In medieval Europe, the Ashk’nazim took upon themselves the minhagh of not bathing, even in cold water, from Rosh Hodhesh until after Tish’a b’Av, not even Likhvodh Shabath . This despite what the Talmud states that bathing for reasons of a misswa rather than pleasure is permitted even on Tish’a b’Av itself . According to Hazal, the fact that the mourning practices associated with this period cannot impinge upon Kvodh Shabath was never in question. Here too, the minhagh contradicts the Halakha (see above, no. 4) and should be ignored .
As for bathing in general during these days, it needs to be stated plainly that minhaghim must be examined in light of current realities. This custom, like many minhaghim, evolved in a specific historical context. In medieval Europe people bathed very infrequently. (According to some monastic records, monks were given a fresh set of clothing once a year, at which time it was recommended that they bathe.) Not bathing for nine days was entirely normal.
Standards of personal hygiene have changed dramatically; nowadays, nearly all people bathe or shower daily. Halakhically, we are all considered ‘Ist’nisim’ (pampered, or delicate) , a definition with very real Halakhic significance .
In addition, there is the question of climate. Many of us live in countries (such as Israel) where the summers are very much hotter than central and northern Europe.
Either of these factors alone would be sufficient reason to discontinue the minhagh. Taken together, it is unreasonable in the extreme to claim that this minhagh should today be considered binding.
 Ashenburg, Katherine. The Dirt on Clean: An Unsanitized History. North Point Press. New York 2008. pp. 69 – 70:
The Roman bath culture died slowly, fizzling out at various times and places in
the waning empire. Ironically, as political and economic troubles made it difficult to maintain the great thermae where the people bathed, bishops, popes
and emperors continued to build and enlarge lavish baths in their residences.
From being a resource for all, the baths declined into an aristocratic preserve.
In the sixth century, Bishop Victor of Ravenna renovated the baths that ad-
joined his episcopal palace with mosaics and marbles. On Tuesday and Friday,
he graciously allowed the lesser clergy to bathe there. As late as the ninth century, Pope
Gregory IV redecorated the splendid baths of the Lateran Palace. (pg. 74)
These privileged enclaves were far from representative of their society as a
whole. In Italy and the western part of the Byzantine Empire in general, the
Lombardic invasions in the mid-sixth century led to a period of confusion and
breakdown. After the Goths disabled the Roman aqueducts in 537, the thermae never recovered. Even if the aqueducts had been repairable, Rome was in too
much disarray to manage the complex operations that furnished the thermae
with water. That still left more than eight hundred balneae in the city, simple
neighbourhood baths of perhaps three rooms, and it is possible that they carried on for some
generations. So, undoubtedly, did smaller baths all over the Byzantine Empire.
Although the invading Germanic tribes admired many Roman institutions,
the baths were not among them. Their preference was for a manly dip in a
stream, at least in warm weather. (The Romans thought the invaders smelled
vile, in part because they dressed their hair with rancid butter.) By the eighth
and ninth centuries, mistrusted by the Christians and neglected by the Germanic conquerors,
the baths in the West had fallen into disrepair and were finally abandoned. Extraordinary
achievements in engineering, architecture, public health and city planning that stretched from
Italy to Britain to North Africa, they mostly lay in ruins for centuries. Some, as at Bath in
England, later returned to full use; others, like the Baths of Caracalla, were never restored.
No bathtubs have survived from the early medieval period, but large wooden
ones were probably used in monasteries and infirmaries. Letters and diaries,
songs and epic poems, chronicles and other official records from the time
rarely refer to people’s dirtiness or cleanliness. We don’t know how and how
often people washed, but given the difficulty of procuring water and an apparent unconcern
about cleanliness, the plausible estimates are “not thoroughly”
and “seldom.” pg 77
 Brooke, Christopher. The Age of the Cloister: The Story of Monastic Life in the Middle Ages, Paulist Press 2002. pg. 79.
 Ashenburg, pg. 78
 Ashenburg, pg 88.
 Archibald, Elizabeth. “Did Knights Have Baths?The Absence of Bathing in Middle English Romance,” Cultural Encounters in the Romance of Medieval England. ed, Saunders, Corinne J. pg. 110.
The passage from Secreta Secretorum can be found here http://www.colourcountry.net/secretum/node52.html at the very end – “& seldome bathynge”
[8a] Thomas Shahan (American Roman Catholic theologian, d. 1932) argued that bathing was common in Europe during the Middle Ages. However, his evidence points to a later period – from the 13th century and onwards. By this time the ancient Minhagei Ashkenaz were deeply rooted in the European kehillos even if the societal norms were indeed beginning to improve. Even according to Shahan’s report the Jews were only allowed to visit the hot baths once a week:
In the Archives for the Study of Austrian History for the year 1859 Zappert has treated at length the question of bathing in the Middle Ages and shown the frequency of the custom. The hot air and vapor baths of the Byzantine peoples were adopted by the Mohammedans, and later on made known to the peoples of Western Europe through the Spanish Arabs and the Crusaders. They were in great demand as a cure of leprosy, and competent authorities state that after the beginning of the thirteenth century there were few large cities in Europe without them. Their statutes are well known to us. The Jews were allowed to visit them once a week. Lepers had separate baths.
Shahan, Thomas J. The middle ages, sketches and fragments, New York, Benziger brothers, 1904. pp. 290 – 291.
That Jews were only allowed to visit the baths once a week is also reported inThe Encyclopædia Britannica: A Dictionary of Arts …, Volume 3; Volume 9 entry “Baths” pg. 437.
 Ashenburg, pp. 81 – 82
[10a] The idea of Tosfot Beitzah is repeated in Meiri of Provence (Beitzah 21b s.v. Amar HaMeiri HaMishna HaReviyit), and Piskei Rid of Italy (Beitzah 21b according to the Gilyon of the Ketav Yad).
Sources of citation from R’ David Bar-Hayim:
 SA, OH 551:16, Rama.
 TB Ta’aniyoth
[13a] Thus this minhagh contradicts both a Mishna and a B’raytha.
 See T’shuvoth MaHarShal 92 who criticizes the minhagh.
 This view was expressed by HaGaon HaRav Yoseph Eliyahu Henkin z’l. See ‘Halachos of The Three Weeks’ by Rabbi Shimon Eider of Lakewood, p. 13.
 See Mishna B’rakhoth 2:6 :
RABBAN GAMALIEL] BATHED ON THE FIRST NIGHT AFTER THE DEATH OF HIS WIFE. HIS DISCIPLES SAID TO HIM: YOU HAVE TAUGHT US, SIR, THAT A MOURNER IS FORBIDDEN TO BATHE. HE REPLIED TO THEM: I AM NOT LIKE OTHER MEN, BEING VERY DELICATE.
Today we are all “very delicate”.