Thanks Joe for those references.
I'm starting to think that this set of words - "Little Jim" - was originally a poem as opposed to a song.
None of the Bodleian broadsides mention an Air or tune.
The only evidence to support its existence as a song originally is that it is in the Roud index (perhaps this refers to Almeda Riddle's musical setting?) and the memory of "E Powell, Coventry" in the poetry link above, who remembers her mother singing it:
"As I read it I was about six or seven years old in a gas-lit room sitting again by the fire-guard of a blazing coal fire, with my back leaning against my Mother's knees. She was putting curling rags in my long hair before taking me to bed. At the same time she was singing this poem. As mother went through the song, my tears began to flow uncontrollably until I had to turn to her saying, 'Mother don't sing any more it's making me cry!'"
Then again, perhaps this was the mother's tune...