Hello, UK resident here.
This is a very common scenario that you’re posing.
There are two factors at play here: Right to work and licencing.
As a UK citizen, you do not have the right to work in the USA. Some schools get around this by offering students paid CFI work under student pay regulations.
These are all time capped and once your visa expires that’s it. Off you go to Europe.
If you get an FAA licence, that’s pretty useless in Europe, you’ll have to convert it to EASA which means taking all the theory exams and re-doing a lot of the flight tests.
So my advice to you is as follows. Train in the type of licence that you want to work in. If you’re planning of living in Europe indefinitely. Get an EASA licence. A lot of the process can be done in the USA to save money but ultimately you need to have right to work.
If you do manage to get a US work permit then by all means go FAA