What would the Old English Wōden look like in Modern English?
Solution 1:
I think it's not possible to give a conclusive answer, per the comments beneath the question:
We have the example wozen → ooze, so maybe Oodin. But I don't know whether the "regular sound changes" were regular enough to know for sure. – Peter Shor
Yes. Most of the sound changes, though regular enough, were dialectal in nature and only affected some words in some places. Then words from some dialects got mixed up with words from others as people moved around and talked about new things in new ways. A lot depends on what the word would be used for; gods get repurposed quite a lot. Look at Pluto, or Thor, or Om. – John Lawler
Basically, the points of possible uncertainty I can think of are:
The presence of the initial semivowel /w/: this was often lost before rounded vowels on the path from Old to Modern English. In addition to the example ooze < OE wāse and wōs that Peter Shor pointed out, there are also two < OE twā, who < OE hwā, sword < OE sweord. But there are also examples of words where w was not lost in similar contexts: swoon < OE geswogen, swoop < OE swāpen, womb from OE wamb, and perhaps most relevant, wood (adj.) from OE wōd (which according to the Oxford English Dictionary is believed to be etymologically related to the name Wōden).
The quality of the vowel in the first syllable. Usually, Old English ō developed to modern English /uː/, as in food. However, in some cases, it became /ʊ/, as in good, or /ʌ/, as in blood. I have a feeling that this generally happened in monosyllables, but I don't really know at all if this is true. We do have /ɛ/ in Wednesday; the OED says that this is "unexplained" and says it might be due to i-mutation (umlaut). It also lists other examples of this form showing up in placenames, but it doesn't indicate that it was ever used independently. Note that in this position, it wouldn't be unusual for the vowel to undergo shortening that wouldn't affect the name as a separate word: the first part of Monday is not pronounced the same as moon, even though etymologically these elements are identical to one another. So even if we consider these names evidence for a form Weden with an e in the first syllable, it might well be pronounced /wiːdn̩/ rather than /wɛdn̩/.
Solution 2:
After all, there is a common English word derived from Woden
Wednesday (IPA wɛnzdeɪ) (Wiktionary)
From Middle English Wednesdai, Wodnesdei, from Old English wōdnesdæġ (“Wednesday”), from a Germanic (compare Proto-Germanic *Wōdanas dagaz)... (ibid; my emphasis)
Another wiktionary page gives an archaic pronunciation of Wednesday as ˈwɛdᵊnzdeɪ', so, absent any other philological evidence, we could do worse than to say Wōden would have developed into *Weden* (ˈwɛdᵊn') in modern English.