"There is some conjecture (esp. Riley 1957) that the Scottish version of 1740 (Child A) is a rewrite of the English broadside, presumably by James Oswald who published the version along with Ramsay in 1740."

I'd be interested to know if anyone has actually seen what it was that Oswald published in 1740 in "A Curious Collection of Scots Tunes". Did he publish the words, or just the melody? Riley says she hadn't seen the book (p. 45) and she relies on Hendren. But Hendren says he hadn't seen any of Oswald's books (p. 71 note 3) and had "pieced together" what he knew of them from the comments of other editors. Bronson mentions it as a possible earlier source for the tune he gives as 84.40, but also says he hadn't seen it. I've found the full title cited as: "A Curious Collection of Scots Tunes for a violin, bass viol. or German flute, with a thorough bass for the harpsichord, and also a sonata of Scots tunes in 3 parts and some masons' songs with the words, for 3 voices, to which is added, a number of the most celebrated Scots tunes set for a violin or German flute", which seems to suggest that it is a collection of instrumental music (apart from the "masons' songs", but I'm not sure how "Barbara Allen" would fit into that category). Oswald's later collection, "The Caledonian Pocket Companion" is certainly just a collection of tunes, and indeed includes "Barbara Allan" (melody only). It can be found online at archive.org, but unfortunately not the "Curious Collection", although I understand it has been reissued recently on CD-ROM.

In any case, Riley's argument for Oswald as the likely author seems rather weak – basically that some comments by his contemporaries suggest that he had some literary skill, therefore he could have written it. Quite possible he could, but given that Allan Ramsay, who certainly published the words (without the tune) in 1740, is a well-known poet, if either of them was the author (which of course is a very big "if"), I would have thought Ramsay rather than Oswald would be the obvious candidate.