I think you'll find a few of them devolved into nursery rhymes - "Hot Cross Buns" is one of them.
I put together a compilation song of several verses I found and a chorus that was a bit different. There are others.
One that comes to mind is "Hokey Pokey" (honeycombe crunch) came from a street cry of an Victorian Itallian icecream seller who shouted "Pocco um Gecca" (?) and a rhyme that went with it was "Hokey Pokey, penny a lump. The more you eat the more you pump"
my research on this (such as it was) indicated that the street sellers had quite lengthy songs in the 17/18th century but as the 19th century progressed they became shorter. Two line stanzas ish.
Now what is that about sound bytes and dumbing down - plus ca change?