Yule started out as a month name. In the Germanic calendar, the two months on either side of each solstice had the same name: so the months on either side of the summer solstice were called litha, or alternatively ‘former litha’ and ‘latter litha’, and the months on either side of the winter solstice were called ‘Yule’ (or ‘former Yule’ and ‘latter Yule’). The year began on the winter solstice, so ‘Yule’ was the first and last month.

In Gothic the month name appears as jiuleis (‘former Yule’ = fruma jiuleis), and in Old English as giuli (Bede, Reckoning of time §15), geola (9th century Martyrology: ‘former Yule’ = ærra geola, ‘latter Yule’ = æftera geola), and other variant spellings.

The Gothic reference is the earliest reference to Yule. It’s a document generally thought to date to the 500s: it is a single page of a Christian liturgical calendar, which was recycled (‘palimpsested’) in the 700s. It’s now in the Biblioteca Ambrosiana, shelfmark MS S 36 sup., better known to scholars of Gothic as ‘codex Ambrosianus A’.

The reference to Yule is usually reported as being in a heading that supposedly reads
naubaimbair . . . fruma jiuleis ·l·
where naubaimbair would mean ‘November’. However, David Landau has shown that ‘November’ is a product of 19th century scholars’ imaginations, and that it just isn’t there in the manuscript. The ‘November’ misreading got people awfully confused for an awfully long time: the pagination shows that this is the last page of the calendar, but people didn’t think fruma jiuleis could possibly be the last month of the year, if it was supposed to correspond to November.

For reference, here’s the complete text of what remains of the Gothic calendar, with an English translation. Note that it’s an older text that still uses the debunked reading naubaimbair, and the translation repeats an older scholar’s misinterpretation of jiuleis as ‘July’. (https://www.roger-pearse.com/weblog/2019/12/16/the-first-mention-of-yule-the-gothic-liturgical-calendar-in-the-codex-argenteus/)

Fruma jiuleis is indeed the last month of the year, and it corresponds tidily to Bede’s giuli. It isn’t November, and it certainly isn’t July.

The text fruma jiuleis ·l· is secure, fortunately (Gothic alphabet: 𐍆𐍂𐌿𐌼𐌰 𐌾𐌹𐌿𐌻𐌴𐌹𐍃 ·𐌻·). It appears to mean ‘first/former Yule, 30’. Landau interprets the numeral as a regnal year (i.e. the 30th year of Theoderic’s reign, which would make it December 522 CE); it seems more likely to me that it sets out in advance the number of days in the month.

The etymology of ‘Yule’ is uncertain. Bede thought it was the name for the solstice itself, which is plausible. Landau, observing that the Gothic calendar is a Christian liturgical calendar, suggests that it’s derived from Jubilee (Hebrew yovel > Greek Iobelaios, Latin Iubilaeus). That can’t be right though: Jorma Koivulehto points out that an Old English variant geohhol and Finnish juhla (a loanword from an Indo-European language) show that there was a labiovelar consonant in the middle of the word. He reconstructs a proto-Germanic root jehwla-/jegwla-. As things stand no one has traced the etymology any further back.
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Festival and customs

There’s just no evidence of Yule prior to the modern era. The clearest evidence of a festival with particular associated customs comes from Old Norse sources of the 9th century onwards, which call it jól or iola. There are frequent references to jól as a way of marking time — ‘so-and-so travelled to such-and-such a place during Yule’, that kind of thing — and multiple sagas have stories of spirits of the dead (draugar) causing trouble or violence during Yule. Only a few sources mention customs associated with a festival. The main ones are:
Hrafnsmál (‘raven’s song’), second half of 800s: stanza 6 refers to the custom of drinking a toast.
Helgakviða Hjörvarðssonar (‘poem of Helgi Hjorvarth’s son’), mostly dating to the 900s: stanza 32 also refers to the custom of drinking a toast, and also that it was a custom to make a vow over a pledging-cup (bragarfull). The prose frame-narrative, which is from a later century, also refers to the custom of a sacred boar that is brought into a feast in procession (sonargöltr).
Christmas adaptations
The evidence we have above about Yule customs is pretty sparse stuff. Evidence linking Christmas customs to Yule is even sparser. Most is conjecture by 19th century scholars.
The date. Bede’s modranicht was on the winter solstice, and 25 December was the traditional date for the solstice in the Julian calendar. But this isn’t a case of borrowing. The assigning of Christmas to 25 December is earlier than any evidence of Yule: the date was set in stone by the 4th century at the very latest, and probably already by the 2nd century.
The boar. It’s possible that the 15th century Boar’s head carol is linked to the Norse custom of a sacred boar. But while there’s a resemblance, there are several centuries and multiple countires separating them. Resemblance is easy: proving a link is another matter. I’d be prepared to accept evidence of continuity linking them together, but I don’t have that evidence at my fingertips. Maybe someone can add something here. The boar does feature in some modern celebrations, but that’s deliberate archaising: it isn’t the same thing as showing continuity between 9th century Scandinavia and 15th century England. Wikipedia suggests that the Christmas ham is a distant descendant of the sacred boar, which is even more tenuous.
The vow. It looks conceivable that the modern custom of new year’s resolutions is a distant descendant of the Yule vow; the solstice is supposed to be the new year in the Germanic calendar, after all. Again, though, it’s very distant and tenuous.
The log. It isn’t a Yule thing, it’s a Christmas thing. The first appearance of a ‘Yule log’ in English is in the 1600s, in a poem by Robert Herrick. He calls it a ‘Christmas log’. It was a Christmas custom that ended up getting the ‘Yule’ label attached to it to make it sound cooler.
The early Germanic calendar
Here’s a fuller account of the calendar that Bede describes, the month names in the Old English Martyrology, and how they stack up against the Gothic calendar. There were a bunch of other Germanic calendars floating around in the mediaeval period, but they’re all later than Bede. Most of the other ones, like Charlemagne’s calendar, are designed mainly to serve as localised versions of the Roman calendar.