We all clean up when speaking to furriners. I do so when abroad in this vast country, and so do many Americans. Also, of course, films and television require cleaned up speech. We tend toward neutral Midwestern, while Brits often migrate in the RP direction. And of course so much Brit TV here is Victorian stuff (starring Judi Dench), which rarely portrays a variety of accents and if so, they're out of date.
I've really had a hard time getting good samples of local British speech on the net, with so much garbage--little YouTube clips that are all noise, someone only says four words, and of course if the video has the word 'accent' in the title, you don't know if it's a real native or some drunk sod taking the mickey out of the neighbors. Or, they're plainly identified as a joke. A promotional piece for some city may or may not be presented by a native--you can't assume it.
When we do get genuine non-RP (Last of the Summer Wine, for example), it isn't usually identified as to its origins unless you go look it up somewhere. So, we don't know one British accent from another through simple lack of information. And they lighten up on the dialect for general distribution as well. Even many Brits would be a bit lost at times listening to some small town Yorkshire folk who still use the old vernacular.
As I said, I clean up too. I'm perfectly capable of graduate-degree level speech, but still say 'good' instead of 'well', etc., when talking to friends and family, and in a terse Hoosier accent when I'm back in Indiana. Chicago is 'Ch'kahgah' in Chicago. If I phoned any of you, it would be in a squeaky clean northern Illinois accent. The Indiany-talk is clear but still throws some folk not used to hearing it as it has a bit of a hard edge.
It must be said that Americans have had their way with the language. And left it smelling that way, too.
Lac lactis in primoris (milk in first).