It’s clearly not uniquely British (as the prior posts show) but I think it is very rarely used in the US. I’ve never heard of it, period, and would have been rather baffled by what “bespoke suit” meant unless it was clear from the context (and the context would have had to have been pretty darn clear on this point!).
Dictionary.com’s entry (based in the random house defintion) for this sense of “bespoke” refers to it as “British”. It defines this sense of the word as relating to either clothes or a tailor, implying that the term is only applied to clothes. However, its sample sentences include examples where the referent is neither clothing nor a tailor.
The Dictionary.com entry based on the World English Dictionary (published by Collins) does not mention whether this sense of the word is chiefly British or not, and it does not limit the “custom-made” sense to clothes (instead, it says this sense is especially used to refer to clothes, a website, a computer program, etc., which strikes me as a rather odd trio to tie off with “etc.").
The MW online dictionary doesn’t say anything about this sense of bespoke being chiefly British or not, either.
So there is some (but not very strong) evidence for it being chiefly, if not uniquely, British.
I wonder (but this is just a WAG, and a particularly wild WAG at that) if bespoke in this sense is at least slightly more commonly used on the east coast than the west coast of the US.