The earliest mention [and no more] is 1549, in The Complaynt of Scotlande [an anonymous book], as being a favourite song of the time--so it's older than that. Next there's mention of a ballad, "A moste Strange weddinge of the ffrogge and the mowse", which was regustered in 1580-- but as far as I know it has disappeared. The earliest text we have is the one I mentioned, in Ravenscroft's book of 1611, Melismata. He gives the tune too. It starts: It was the Frogge in the well,

Humble-dum, humble-dum,

And the merrie Mouse in the Mill,

tweedle, tweedle twino.

The burden in between the story lines has to be an imitation of the sound of a spinning wheel--Mrs, Mouse is spinning when Froggy arrives. Same goes for the more modern versions-- "ahum" or however you want to spell it, while the "twino" seems to spawn "kimo" as in the minstrel song ["coino" is found in the 18th century], and there's quite a few variations. One reference: The Oxford Dict. of Nursery Rhymes (by the Opies), which gives more references. The American scholar Kittredge has a good treatment of the song in the Journal of American Folklore way back in the 20s, I think.