According to my uncle's notes:

"The Opies in their Oxford Dictionary of Nursery Rhymes, says this (song) may be traced back four centuries, since it is clearly referred to in The Complaynt of Scotland (1549). The oldest text and tune now known to exist are in Thomas Ravenscroft's Melismata (1611)."

My uncle added a final verse to his cheerful American rendition:

Then Frog and Mouse lived happily, mm-hmmm.
Then Frog and Mouse lived happily, mm-hmmm.
Frog and Mouse lived happily,
And they were blessed with children three,
Mm-hmmm, mm-hmmm, mm-hmmm.

And one did croak and two did squeak, mm-hmmm.
And one did croak and two did squeak, mm-hmmm.
One did croak and two did squeak,
And they had long tails and webbed feet,
Mm-hmmm, mm-hmmm, mm-hmmm.*

A celebration of diversity! Of course, we kids prefered the verses with the mouse and frog getting gobbled up.

* Verse by Richard Dyer-Bennet