That's a promising idea Jim, but I've just looked at the books listed in the Roud index that have music (ignoring SMM and Dick) and they all showed the original tune. (Lyric Gems of Scotland 2 (1856?), Scottish Minstrel (Valuable Selection, c1813), Smith, Scotish Minstrel (1st edn. (1820-24)).

That still leaves the possibilities of sheet music, a popular singer or a very early popular recording.