Okay, Malcolm, I've taken the bait.
The one you don't want 'Roud 847' 'I don't mind if I do' is common in British oral tradition and on broadsides from about 1850 onwards.
The previously posted 'Joe Muggins's Ass' seems to be based on 'JM's Donkey' which is, as has been noted, a parody of 'Lord Lovell' Burlesques and Coster versions of broadside ballads were very popular in the early nineteenth century reaching a climax with the performances of Sam Cowell and Fred Robson in the 1850s. In some cases the words were very little altered and the humour was injected by putting on an exaggerated coster's accent and lots of facial grimaces whilst wearing the appropriate costume. For instance it didn't take much doing to burlesque Lord Lovell, the burlesque version is almost word for word the traditional version. Other well-known burlesques were Billy Taylor (William Taylor), Villikins and His Dinah (William and Dinah) Molly the Betrayed or the Fog-bound Vessel (The Cruel-ship's Carpenter/ Daemon Lover), Barbara Allen almost unchanged and many more to a greater or lesser degree based on traditional ballads (All around my hat which was parodied and burlesqued mercilessly).

As for origins very few of these have been attributed accurately. They are often attributed to Sam Cowell as they appeared in his songbooks, but some of them were on the go before he was born.

The earliest version of JM's Donkey I have been able to find off hand is Hodges/Disley which puts it at mid nineteenth century, but I've a feeling I've seen it in The Universal Songster which would date it before c1825.

SteveG