The character in question lives in America and was attending Harvard, and I want him to transfer to Cambridge in England. Is this possible? Some information that may or may not matter: this character is studying law, and it's the mid 90's.
There are certainly american students at Cambridge so there shouldn't be a problem if his grades are good and his parents can afford the fees http://www.undergraduate.study.cam.ac.uk/international-students To clarify an international law student will pay circa 20k pa in university fees, another 8k pa or so in college fees and need roughly 10k in living expenses at minimum - so thats £38k per annum in all that's $51,280 at the current exchange rate (obvs thats at todays prices 20 years ago it was less, but on the otherhand wages were lower too) https://www.undergraduate.study.cam.ac.uk/international-students/fees#college
My daughter's college has a program where you do your first two years in the U.S. and the last two years abroad at a "sister school." I don't think it's that uncommon a program for private universities in the U.S.