"Off She Goes" is near as dammit the tune that's traditionally used for the nursery rhyme "Humpty Dumpty Sat on the Wall"   And it's cited as the tune for "Walker Pits", printed in Allan's Tyneside Songs (published 1862 - reprinted with introduction by Dave Harker 1972), though the text was first printed in Bell's Northern Rhymes of 1812.

"Walker Pits" has the same chorus as currently sung with "Byker Hill", plus two of the same (well almost)verses ("If I had another penny...etc" and "When aw cam to Walker wark/ Aw had ne coat, nor ne pit sark...etc). Other verses have since been added, some of them borrowed from another Tyneside song "My Dearie Sits Ower Late Up"

The 9/8 tune which Martin Carthy and Dave Swarbrick applied to these words is very similar to the Northumbrian Pipe tune "Durrington Lads" - perhaps they got it from the High Level Ranters? Or maybe from Bert Lloyd?

The marching tune used for the same words by The Young Tradition is also the tune of the American spiritual "Where are the Hebrew Children?" On another thread somewhere on this site there is some discussion of whether the YT got this tune from Sandy Paton, or from Red Sullivan. (Worth looking up?)

And in Byker, at the Cumberland Arms sing-around (first Wednesday of every month), the 'original' words of this song are still sung, but to a completely different tune! You can hear it sung by local group Kiddar's Luck here: https://www.youtube.com/watch?v=OWqIoJp_hMQ

(Though they do it a bit faster than it's usually sung in the Cumberland.)

And that's enough for now - Wassail!