Everyone gets their version of their native language exactly the same way, by growing up among people who speak that way.
Actually, there is no such thing as A British accent, there are lots of them, depending on where you grew up and were educated.
Few people who learn another language after their formative years can speak without an accent. For most, here are sounds they can't make, there are ways of talking that are much more natural to them.
Everyone speaks with SOME accent -- you just don't think of the way you talk as having an accent, but people who talk differently would describe you as having an accent.
There is no one American accent: people from Georgia don't sound the same as people from Washington, Texas, or New York (to name just a few).
A lot of people can imitate other people's accents, but there's usually a few sounds that give them away.
So, for instance, in the TV series (a British show) As Time Goes By, there is a recurring character who's American.
The actor's accent is pretty good on the whole, but to this American's ears, he doesn't sound QUITE American all the time.
(For fans, I'm referring to Mike Barbosa -- as I say, most of what he says is spot on, but now and then I can tell he's a Brit imitating American speech.)
I bet when you and your friend try to talk with British accents, real Brits would say you're getting it wrong.