I've heard it sung to a common but very nice minor tune used in Britain for a number of songs including 'Manchester Angel', 'The Lass of Swansea Town' (Phil Tanner). Martin Wyndham Read used to sing it to this tune.
Whether that relates to versions as collected I don't know. The only
Irish Molly-O! tune I know is a jolly affair. It features in almost all of the Australian song books I have and gets a mention in Lloyd's Folk Song in England.