I'm sceptical. If that was Alexander Campbell Mackenzie (art music composer and the biggest cheese in the Scottish musical scene in his time) he'd have been quite capable of claiming an old folk song as his own.

And the tune is (nowadays at least) used for the song family "Oh that I had ne'er been married"/"Crowdie thrice a day", which goes back to about 1700. (I've transcribed old versions of that but I can't find them right now and can't remember what they sounded like).