If there's a confusion between Calton, Carlton and so on, it's down to revival singers rather than the tradition. Many of them will have learned the song from other people's records and just guessed at the spelling. Most if not all of these will derive (through various intermediaries) from the set published and recorded by Ewan MacColl, one of only two that mentions Calton; and the only one of those with a tune.

'Nancy Whisky' seems to have started out as a broadside song; there are two editions at Bodleian Library Broadside Ballads, one of which is set in London, while the other (printed in York) names no particular place. See  Nancy Whisky.

There are other localisations; some of the Scottish versions in Greig-Duncan mention Dublin (probably deriving from another broadside localisation: see http://www.nls.uk/broadsides/broadside.cfm/id/16420/ ), and Sam Henry got one in which the weaver belonged to 'Long Cookstown'. Willie Scott mentioned Stewarton. It has also turned up in the south of England.

Almost certainly nothing at all to do with Calton to begin with, then; but taken up there at some point and adapted to localise it. The historical background of Calton itself is interesting, but in the circumstances only peripheral to the history of the song. MacColl got it from Hughie Martin of Shettlestone in Glasgow, who, it seems, reckoned that MacColl's father had written the tune, "because he didna' tak' to the ither yin". (Singing Island, 1960, no. 36: 36, 41 and 110).

See several previous discussions here, some of which include further detail and the usual comments on how 'Calton' should be spelled.