Unlike for the tune in Dan Milner's book there is no doubt about a connection between the song Eric mentions and Jacobitism.
Simon Fraser's Collection includes a tune (#225) titled "The Rising of the Year 1715" for which Fraser states:
"The chorus of this air and its name are well known to allude to the rising of the year 1715...this little ancient air embraces the subject of two favourite Scots tunes, which seem to have been built upon it, viz. 'O'er the Muir among the Heather' and 'Peggy now the King 's come'... "

'O'er the Muir...' is the song you mention (Scots Musical Museum #328) while 'Peggy...' is SMM#239 for which a Jacobite version also exists ("...An somebodie were come again, then somebodie maun cross the main...")

The (capital) information provided by Becky hints at an old (ultimately?) Irish song and not a modern one as supposed as yet in the discussion. It makes the hypothesis that this tune -or a variation of it- could be the same as in "Mackintosh" at least plausible.