To update this query: Although nobody responded on Mudcat I did hear back from several folks from the Cecil Sharp House Ballad Discussion Group, who concur that the Bronson citation above does NOT correspond to the tune in question, nor is any other Bronson tune the same as Sarah Makem's.

I have three oral sources for this tune, all from Northern Ireland. The two I mentioned above, and also Nielé Mhiéi Hiúdái from Ranafast, Donegal. I don't have a recording of her, but her version was passed on to me by Dáithí Sproule, and the same version was also recorded by her niece Tríona Ní Dhomhnaill.

Mary Humphreys pointed out to me a Wikipedia entry stating that Tríona's aunt Nielé contributed almost 300 folk songs to the National Folklore Collection in University College Dublin! After the holidays I'll contact them to find out if her version of Barbra Allen is in their collection.

In summary, either nobody has investigated the origin and history of this lovely tune in a scholarly way, or else their work is quite hard to find.

Best to all,
Pamela