In answer to the original question, in my experience these sort of songs are made up from a few well-known floaters that can be added to ad-infinitum. The usual tune, yes, is Villikins, but a wide variety of choruses is used, so as for a title simply make up your own.
'Too-ra-loo'
(first line)
'A rolling stone gathers no moss' (my version)
'My Uncle Jim'