Apologies for the delay but I've finally read the section on Chappell in Karen McAulay's thesis - many thanks to Jack for the link.

IMO she doesn't paint Chappell in a negative light at all, and doesn't consider him to have been anti-Scots. He did have a mission to dispel the myth that England had no national music, and he did make some mistakes, but she doesn't seem to think he had an "agenda" in the sense of the word that would imply he would be willing to intentionally twist the truth or mislead people in order to promote his aims. I think she considers him to have made serious efforts to be fair. Many of his Scottish contemporaries did consider him to have been anti-Scottish but Karen seems to think that this says more about their Scottish nationalism and their dislike of the idea that some tunes that were then considered to have been Scottish might actually have originated in England than it does about Chappell.

She points out that he had planned to write a collection of Scots tunes, and although he never got round to it he did do some work on this. She ends her section on Chappell by writing:

William Chappell clearly enjoyed a continued interest in Scottish music, and was meticulous in his efforts to set the record straight as he perceived it. Admittedly, some would argue that his theories might have been misguided. Yet arguably it was not this, so much as the sensitivity of nationalist epistemologies that caused such upset.

If we accept that he was basically honest and meticulous and that he tried to be fair, which Karen clearly believes to be the case, then I don't see why we can't accept that when he wrote: "The tune is the usual traditional version", he honestly believed that to be true. Maybe he did get the transcription he used from Rimbault but I don't think that proves that he didn't also hear it sung orally as a traditional tune.

The fact that he didn't quote his source for the tune seems to me to have been unfairly held against him in this thread, given that it applies equally to both Rimbault and to Motherwell.

Steve says the tune has a similarity to Tudor pieces in style, and if true, that would back up the idea that it is possible that it goes back to the origins of the song, which were indeed in the Tudor period. (It was first quoted from in a 1611 play so it was clearly already very well known by then, which means it was probably dates back at least to the Elizabethan era).

So I think it is fair to say that it is a traditional tune that the song was probably sung to in both England and Scotland in much of the 19th century and quite likely much earlier; and that it is possible that it is as old as the song itslef.

I also think it's highly likely that it originated in the Durham area, given the protagonists' names; and so I'm going to sing it in a Northumberland accent.

Many thanks for everyone's help with this, it's been fascinating.

Dave