The oldest known version of the tune is quite definitely Scottish, and its association with Burns's text is what made it familiar worldwide. Its earliest known text, I think, is "The Brown Dairymaid", a Gaelic song about the 1745 rising (and written very soon after it) by Alasdair MacMaster Alasdair, which is what Burns heard on his trip to Inverness. But it doesn't sound very Scottish, and my guess is that it started out as one of the Italian-Baroque-influenced tunes created in England in the late 17th century and labelled as "Scotch". You can imagine it fitting pretty well into a Baroque violin sonata as a siciliano-type slow movement.