Thanks for all the info everyone...

The tune I'm after is definitely the one that goes with "Little Jim" aka "The Collier Boy" aka "The Collier's Dying Child". The parody conforms with the Bodleian broadsides very closely.

Unfortunately there are no tunes! (I know the Bodleian collection contains only a little music).

The Canadian reference (thanks for this also Malcolm) doesn't have a tune either - but perhaps Almeda Riddle's version does - is there anyone with access to this who would be kind enough to check?