I'd guess there were two different last-lines current:

1. Jenny get your hoe-cakes -- revised by Britishers as oat-cakes, and based in the minstrel chorus -- nothing 'obscene' about it.
2. 'Dirty' version, 'Jenny keep your ---hole warm', which can be understood as a parody of #1.

Hugill knew of #2, and was being coy about it -- perhaps leading Revivalists to guess (and leading to "get your oats my son". Short may have known #2, but was too much of a gentleman to offer that, and offered the 'original' instead.

I realize this scenario does not explain the 'long tailed Black man' -- the reference may have been too opaque to raise any eyebrows. Although I have a 'dirty' interpretation of it, I'm by no means sure of that. If I wasn't told the song was supposed to be obscene, or if I was presented the song with "hoecakes", I would probably rationalize the long tailed black man in some other way.

I think all these things are true:
--Sharp did not bowdlerize, i.e. there was a fairly harmless version
--There was a bawdy version, too
--The song *has* been bowdlerized in some/various cases, too!
--The song has been re-bawdified, too!