Interesting sozobe. Have you ever tried to converse ASL with foreigners?
I understand that even the British have different signs, even though it's
the same language.
Not sure what you mean by "the same language," -- even though Americans and the British both speak English?
Anyway, yep, British Sign Language is completely different from American Sign Language. I just had to rely on lipreading in England. That's true of almost all foreign sign languages, but French Sign Language is the exception. ASL was based on FSL (French educators came to America and taught American deaf children their sign language) so there is approximately the same relationship as American English to British English.
That means that I can communicate just fine, in sign language, with deaf French people. It's very cool. (I've gotten some great tours in Paris from deaf French people I happened to run into.)