It goes both ways - there are American actors who portray British characters, and even I will spot it, and it’s like someone smashed a plate over my head.But I have heard the Brits american accents slip at times and I find that sort of thing a bit annoying.
The one US English word that seems to trip British actors up…figure. We say “fig-yer,” and our British friends largely say “figger,” and it’s rather alarming to hear someone on a TV western say, “I reckon I’m a-gonna mosey on down to the saloon right quick-like. Miss Kitty will be lookin’ mighty good to this ol’ cowpoke, I figger.”