What I can tell you:
* The two official spellings are Gwylim and Gwilym, though Gwylim appears more frequently.
* The University has it's publisher,
'The University of Gwylim Press', which shouldn't be too confused with the University itself.
* Erer Darothil is mentioned as have been a Proffesor/Teacher of Illusion there in the book,
http://www.imperial-library.info/mwbooks/silence.shtml, however he also could merely have been fictional.
* Tjurhane Fyrre, a Valenwood-Ayleid, is mentioned as being
'one of the finest sages of the University of Gwilym', in the book http://www.imperial-library.info/obbooks/wild_elves.shtml, and also wrote a book called
Nature of Ayleidic Poesy (published via The Univerisity of Gwylim Press).
* In the second of
http://www.imperial-library.info/obscure_text/Xal_Gosleigh_Letters.shtml, it's implied that both Xal and Gosleigh attended the University, aswell as Divayth Fyr.
* From the http://www.imperial-library.info/obscure_text/forum.shtml
"I would not take the ravings of this "Michael" too seriously. He has never been published in the Journal of Ancient Tamrielic History, and was dismissed from the his position at the University of Gwylim for persisting in reading aloud portions of the long-debunked "Monkey Truth" during faculty meetings." (apparently the Univeristy doesn't hold MK in as high regard as this forum does )