Righto, Peter. We do notice the differences first, as there are several things that almost all British accents have in common that American accents do not, and vice versa. But if you were to make a video consisting of brief conversations with people from a couple of dozen British regions, we'd hear the differences. One problem is that we can't categorize the accents in our minds because we hear actors, news readers, and people being interviewed, usually without knowing where they're from. Samantha Morton is very Nottingham, and Michael Palin retains much of his Yorkshire accent, but without knowing that... I've heard English accents so slight they sound like expats who moved to the US when they were three, and others so thick you can barely tell what language they're speaking. But these are exceptional.
A close listening to American accents reveals much, though. Wealthy Americans sound like English southerners, while the yokels twang away like Fenlanders, who have a collection of diverse accents you never hear over here, and from what I can tell not much in England either. An attempt to get video samples of Fenlander accents in Google gets you nowt.
