I think you got it somewhat topsy-turvy there Strawie, old chap.
Americans would never call a fanny-pack a "bum bag," since "bum" in US idiom means a homeless person or a panhandler. "Bum" as meaning a person's rear end is exclusively non-American usage. The word of choice here is "butt," "bottom" or "ass." "Bag" also means something that you carry, with handles.
In fact, Americans call it a "fanny-pack" because a "fanny" is also (less-popular) slang for the human bottom, whereas in EngIand I hear it means something else (ie, Strawie's favorite word).
Also, we say "tights" as well as "pantyhose." I hear in the Uk "tights" are opaque and "pantyhose" are transparent--here they both mean either.
And I've never heard an American call a postman a "letter carrier." We call them "mailmen."