No previous discussion as such, it turns out; just a comment and a reply from me. This is number 1047 in the Roud Folk Song Index, with examples cited chiefly from England and Scotland (earliest appearance in print seems to be Chambers, Popular Rhymes of Scotland (1870 edn, 140-142: I presume the Edinburgh example mentioned above).

Also called Jinny Jo, Jilly Joe etc; Ann Gilchrist discusses it briefly in the Journal of the Folk Song Society (vol VI issue 22, 1919, 84-85) and demurs from Lady Gomme's idea that it is a "drama of domestic occupations and village custom" and Newall's (mentioned above) suggesting instead that earlier forms of the song may have been more narrative, including an account of the girl's death; and pointing out that Chambers' set makes it clear that Jenny is dead all the time and the reasons given why she can't be seen are delaying tactics such as occur in Edward and other ballads. Speculation, but interesting.

Related, though lacking the final dénoument and reduced to a "choosing colours" dialogue, is Jenny Jenkins (Roud 731), which has been discussed here a couple of times.