Is it possible that the conflation of Hares on the Mountain is simply the result of format? Its format is:
Line
Line repeated
New line
Nonsense refrain line.

This is a very simple format, and one might expect it to be very common; but after a lot of head scratching I could only come up with three: Hares on the Mountain, Knife in the Window/Sally My Dear and some versions of Bonny Annie/Banks of Green Willow. Once two of these songs are common in the same area (East Anglia in the case of the first two) a distinctive format can easily lead to amalgamation.