> Riley did not have Oswald's version either. It is curious that both came out in 1740 which makes it clear that one was taken from the other.

Perhaps not so curious. Edinburgh wasn't such a big place in 1740, and Ramsay and Oswald must have known each other. (Ramsay is generally considered to be the author of an anonymous poem lamenting the loss to Edinburgh musical life when Oswald left for London.)

Like Lighter, I would guess that Oswald's book is most likely music only, and Ramsay's is certainly words only. If Oswald did indeed publish the "Barbara Allan" tune in 1740, around the same time that Ramsay published the words, one explanation would be that the song was already circulating in something like the form that Ramsay published it, and was sung to that tune. But I suppose the possibility that Ramsay wrote the words and Oswald composed a suitable tune, or borrowed an existing tune, isn't excluded.