Rachel Griffiths, who is Australian, has a flawless American accent on "Six Feet Under" -- a great, high-end melodrama on our cable channel, HBO. I've heard her interviewed on talk shows, and her Australian accent is quite pronounced (to an American ear). Watching her on "Six Feet Under," I'd never know she wasn't American.
When we say "British accent" in America, I think we usually mean the type of accent Sean Connery used in his James Bond movies or maybe Alistair Cooke on "Masterpiece Theatre." I believe Connery grew up in working class neighborhood in Edinburgh, so he clearly didn't learn that accent at home! The same is true of Richard Burton (another identifier for Americans), who grew up Welsh, I believe.
Most Americans have a very limited understanding of the variety of accents throughout Great Britain. We understand the Scots and Irish (though most of us cannot tell them apart), and cockney (but only from "My Fair Lady"), and that's it. We tend to associated accents with individuals, like the characters on "Keeping up Appearances," for example, or Benny Hill!
Accents like those used in Newcastle or Liverpool, just sound equally incomprehensible to our ears. Some of us may get a sense that Prince Charles speaks with a somewhat pinched accent, but I think we assume it's an affectation rather than a class identifier.
Here in America, those of us who grew up in the North have difficulty differentiating Souther accents, though many Southerners can differentiate each other by state! I can hear the difference between a Texan twang and a Virginian drawl, I think, but that's only because of the extended exposure I've had in recent years to the Texan idiom thanks to our current President.
I wasn't aware that the British dislike the American accent as a rule. Is it possibly just part of a parcel of things the British don't like about America (heaven knows there is plenty to put in the parcel)?
For my part, being American, I prefer the "standard" British accent to American for all things except (shall I say discreetly) intimate conversation. A British accent is profoundly unsexy to me. Sexual slang sound simply wrong with a British accent. I've heard plenty of British swearing and foul language, but it just doesn't seem to have the punch it does in American. I think we're much better at swearing and dirty bedroom talk! What a distinction!

But then it might entirely be my
personal bias.