Obviously there are a lot of subjective words in the question. There are dialects of British English that don't sound distinguished at all (Cockney). Also, what sounds distinguished is somewhat (though not entirely) subjective.
However, I don't know how else to ask this. Sometimes I have heard an American speak and thought, "wow, he sounds just as distinguished as a British speaker". But I don't know where that dialect would come from. More often than not, though, I hear an American speak and think, "wow, why don't we sound as distinguished as our British counterparts?"
For what it's worth, I think what makes a dialect sound distinguished is that it sounds "educated, upper class, articulate". At least that's my guess.
Ignoring the subjectivity of the words composing the question, where would you find American dialects that sounded distinguished?