A while back there was a documentary on TV(?) about The Critics Group and EMcC...This included Charles Parker(?) singing "The Strawberry Roan". Unfortunately, his upper class accent did nothing for the song at all. This may be because he was trying to sing an american song.His plummy voice rendition almost made me fall off my stool laughing!
I would say therefore that it depends upon how 'posh/upper class/Eton/Harrow/Roedean'your speaking accent is and that you should perhaps, if necessary, choose your material accordingly. Otherwise....just sing and do the songs justice.