I know this is a very old thread, but anyway, if there is anyone still out there, the song being recalled is "Sing Me a Hebridean Song" by Rae Elrick (real name Archibald Hyslop, an Inspector from the Scottish Education Department who had an alternative career writing comic plays for BBC Radio and comic songs often performed by the Scots comedian Harry Gordon). It was correctly recalled as being performed by Duncan Macrae on the BBC Hogmanay programme from 1962. I know this for sure as I have a 50 year old reel-to-reel audio recording of that very show with Macrae singing it.