"us" for "me" and "f" for "th" in the UK

Why do the Brits use "us" instead of "me" in informal speech? ("Give
us a go" = "Let me have a turn.") Are there any historical reasons?
I'd expect to find a British dialect that uses "her" for "their,"
since this has a clear OE precedent, but "us" for "me"? It doesn't
make sense.

Ditto "f" for "th" (especially in London).

Thanks,
Marc
.