@MontereyJack,
Your term 'British' accent is all encompassing. It includes Scotland, England and Wales, all the accents are very different, and that does not include all the regional varieties. What you presume to be a 'British' accent is most likely a Middle Class Home Counties type of pronounciation. That makes it just one of many.
This is Ross Noble, he's got a Geordie accent.
This is the Scouse ABC
They're both British accents, but I bet when you hear the term you don't think of either of them.