Thanks for pointing to that; I had overlooked the Douce reference. Chappell considered the ballad to be the work of Elderton, while Simpson thinks it an imitation. I don't know if the Bodleian attribution is based on more recent information. Simpson places the Douce print as one of those following the re-registration of the ballad in 1675.

I also missed his reference to a traditional survival of the tune in Wales. Four versions appeared in The Welsh Folk-Song Journal (vol 2 part 3 no 107, 1922) set to Welsh words but retaining the tune-title Susanna. One is reproduced in The Journal of the English Folk Dance and Song Society (vol III, no 3, 1938, pp 170-1) with a verse of Susanna set to it; in Wales it had been used for a Christmas song, Wel dyma'r hynod wyl Nadolig.