I have 3 different songs with this title, all 19thc but none of them are versions of your song. And, guest, at least one of them is worse than this one.

1) Harry Clifton's song

2) found in American collections such as Morris and Hubbard.

3) A local Hull broadside pr about 1870. (Actually it's about a stepmother!)

Incidentally until recently cheap M-I-L jokes were part of the stock in trade of British stand-up comics.