There is no such thing as "a British accent". You realise when you say that you are encompassing three separate countries as well as literally hundreds of very different regional dialects, right. VERY different. So different that there are many other places in England alone where I almost cannot understand what people are saying.
It also seems to refer to what we'd call The Queens English accent, which extremely few people actually have. I guess because when Americans on TV play someone from here they always do that accent? Nobody talks like that! Apart from, annoyingly, English people in some movies :/
Not having a go, it just irks me.
In your post, you used the indefinite article for, "A British Accent", suggesting just one among who knows how many accents and dialects. I do believe everyone from Britain has an accent that is also from Britain, a British accent you could say. You make all kinds of assumptions--that they don't know there are very different accents and dialects(and greasy sour basilects), and vernaculars, that they are unaware of of the different countries in the United Kingdom. Not everyone has a sensitive ear and if you are not of a place, you often do not pick up on differences, that, clear as they are to a local, are very subtle for a foreign ear. Just because someone can't pick out an Irish from a Scottish, doesn't mean they are unaware of the different places. I usually can't really tell New Zealanders from Australians.
Also, there is nothing special or unique about England and its famous disparate varieties of english. Do you think people from Georgia, Boston, Louisiana, LA, New York, Kentucky, North Dakota all have the same accent and no regional particulars of language? No, of course you don't. Similarly, I would guess you can identify, by speech, which borough of New York someone is from? I used to work for a market research firm, and we would get American contracts sometimes, and there are great swaths of that country populated with people who, whatever it is they speak, it isn't something I could understand, even though we both identified as english speaking people.
Where I am from, in Canada, if you drive for about forty minutes out, you get a very different drawly kind of accent. We also have, like anywhere else, sociolects and ethnolects, making any sizable city an endless quilt of variation in accents. We also have two official languages (and there are many varieties of french whose speakers cannot understand each other, like habitants, or Montrealers with their Joual, or the snobs of Quebec City who speak a french that is actually intelligble, or the snarly slobber of Franco-Ontarien.) Besides our two official languages we have--I don't know how many--at least five major indigenous languages, all with very distinctive regional dialects, and of course we attract between 300 000-400 000 immigrants a year who bring their languages, their accents, their slang, curses and food jargon that gets absorbed by the utilitarian english language.
I have three siblings. We have quite a span of age between us which means we each spent our formative years in not quite the same circumstances. Two of my sisters have distinctly separate Anglo-Quebecker accents, my youngest sister has a lazy rural drawl, and people tell me I speak with virtually no accent--whatever that means. That is how stratified and acute accents can be anywhere in the world, not just in jolly old England.
So, all the varied peoples of North America(who you sludged all together under one homogenous monicker, "Americans"--you know, like you accused them of doing with the peoples of the United Kingdom) and their vast sprawling countries are, to one degree or another, aware of just how varied accents can be.
Lastly, I think we can all agree, that Dick Van Dyke sports the most enchanting and endearing English accent ever, as lovable scruffy Bert in that timeless classic, Mary Poppins.