I figure most people from outside Great Britain tended to be exposed to or grow up with material that presents the British as speaking with the English Received Pronunciation. Oh, sure, it's as sophisticated and sexy as they come (Stephen Fry and David Attenborough's voices are liquid butter I could guzzle down all day, every day). As a kid, I also used to think that Barry Ingham's take on Basil of Baker Street was absolutely awesome, too.
Then there's the actual reality of Great Britain's treasure trove of accents. Like anywhere else in the world, really, you can't pop up in England and say "Boy, these people do sound British!". There's Scouse, Estuary, Brummie, Cockney, Pikey, lasting traces of Received Pronunciation (mostly on TV or as spoken by the cultural elite), plus whatever else I might be forgetting.
To answer the OP's question, yes, the cliché British accent is very sophisticated and cool. I just haven't heard anybody from across the pond use it in a realistic manner, unfortunately.
The same goes with Québec, really. You'll constantly hear that us French Canadians have a specific accent that Americans and French both like, but they fail to consider that Québec isn't quite as Cajun as New Brunswick or Prince Edward Island's French speakers can tend to sound. Then there's the French Canadian diaspora in the Prairies and in British Columbia that all have their own accents and jargons, and I'm pretty sure you could go all the way up to Yellowknife and find the occasional speaker of what's technically French, but that nobody from Montreal could conceivably understand.