When I listen to the Beatles' "LET IT BE", I can distinctly hear that the first T is pronounced like Polish or Russian or Spanish "R", only more quickly and softer. But when I watch "The Negative Reaction" episode from COLUMBO TV series, I can distinctly hear that when Lorna says "Mrs Peterson", her "T" is pronounced like Polish or Russian or Spanish "D", not "R". I asked my Tennessee friend about it and he says that he hears no differences. But I hear the differences! Even that edition of Advanced Oxford dictionary that I own says that "it is like a brief /d/ or the r-sound...". That is the dictionary implies that there can be two versions of flap-t. But all Canadian and American students I ask about it say that they see no difference. Can native British English-speakers tell me if I am right when saying that I see the differences in pronouncing this FLAP T.
As per the comments from Janus Bahs Jacquet and tchrist, there is bound to be a range of variations in the way flapped T is pronounced, which could be regional, social or idiosyncratic to a particular speaker.
In terms of standard forms of English, flapped-T is primarily an American phenomenon (it also occurs in Australian English and some regional accents of England and Ireland). It must be borne in mind that, whilst The Beatles (Paul McCartney, at least) and Abba undoubtedly sang with American-influenced pronunciation, neither group was from the USA, so their pronunciation was almost certainly influenced to some extent by their native accent/language.