I don't speak with a British accent, even though I wouldn't mind having one. I still speak with my native accent, although I tend to use British words and phrases. Some people pick them up quickly. I've also noticed some North Americans seem a bit self conscious of their British accent (if they've picked one up) when they meet me and hear my accent. Usually they put on a very broad version of the accent from where they came soon into our conversation.
That being said, when I hear a North American speak who hasn't been here long or is a tourist (and sorry, this includes Canadians), their accents seem so strong and foreign to me. Sometimes when watching TV with a lot of American/Canadian accents it sounds fake to me. I don't watch too much TV, and when I do, a good portion of it is British so that might have something to do with it as well. I sort of have an idea of what a lot of North American accents sound like to British people.
My husband also says that Americans accents soften over time even if they don't pick up a British accent. He said it's not always noticeable until you compare it to someone who hasn't been in the UK very long.