According to the 2007 survey of Harmon, Diamond, and Taylor, in 1985 "Fermat’s Last Theorem was still not known to be true for an infinite set of prime exponents". [Note: I found this paper using ChatGPT. Normally I would have found it using a simple Google search, but with the advent of AI searching that way has become much worse!]
I suspect that the later FLT developments pre-Wiles also did not establish infinitely many prime exponent cases, but those more familiar with the Taniyama-Shimura connection can weigh in.