It seems "What for?" is BrE. And "For what?" AmE. But it may be that both expressions are interchangeable in BrE and AmE.
So what is your question? :)
I wouldn't say one is BrE and one is AmE. Usage depends on the context.
"Please jump up and down three times".
That could easily be heard in either country.
"I am going to make you pay".
Would also be common in both places.
You might occasionally use "What for?" in the second example, but I wouldn't expect anyone to say "For what?" in reponse to the first question. Either in Britain or America.