What is a rough order of magnitude estimate? $$ $$ There is a thread on Meta about this question, http://tea.mathoverflow.net/discussion/567/rapid-closing-of-questions/#Item_0
The methods introduced by Wiles, and by Taylor and Wiles, in the two papers that proved FLT, as well as the methods introduced by Ribet in his earlier paper reducing FLT to Shimura--Taniyama, are at the heart of much modern work in algebraic number theory and automorphic forms, such as (in addition to the proofs of Shimura--Taniyama and FLT) the proofs of Serre's conjectures and the Sato--Tate conjecture.
Conferences/workshops in these fields typically attract on the order of magnitude of 100 or so particants, which gives you some sense of the number of students/researchers thinking about these questions: its in the tens or hundreds, but probably not in the thousands. Of course, not all these people know all the details, but the people at the top of the field surely do. (Of course, there is a question of what "understand" means exactly. I don't know how many people have both carefully studied all the details of the trace formula arguments that underly Jacquet--Langlands, Langlands--Tunnell, and base-change, and also carefully studied the details of the p-adic Hodge theory and other arithmetic geometry that is used in the arguments. But certainly the top people do understand the significance of these techinques, and are fluent in their use and application, and understand both the overall structure and strategy, as well the technical details, of the proof of FLT itself (and of various more recent related results).
Finally, let me note that the best evidence for the final claim of the previous paragraph is that this is currently an extremely vibrant area of research, which has progressed at a rapid clip over the last ten years or so. (The reason for this being that people have not only assimilated the arguments of Wiles/Taylor--Wiles but have improved upon them and pushed them further.)