Alas, I'm in a state of enormous ignorance here, which is why I thought I'd cast a line for further clues. If there are any other collected copies of this carol in Liverpool, they're not in the Roud index and I haven't tracked them down elsewhere. The manuscript sadly doesn't give any helpful information about where in Liverpool it was from, source, etc.

It appears alongside a version of 'Christ was born in Bethlehem' which also lacks information about the source, and just bears the note "as sung in Liverpool".