http://chroniclingamerica.loc.gov/lccn/sn88064479/1861-11-25/ed-1/seq-4/

There seems to be another The Tailor's Courtship from 1861. It's in the upper left hand part of the page page and you'll have to enlarge the image to read the lyrics.

That aside, I have not found a melody although there was reference to a melody at a site I encountered. The 'nursery rhyme' you quoted appears in Google books frequently.