There’s mention of the The Marriage of the Frog and the Mouse in The Complaynt of Scotland (1549) but the earliest known printed version is in Thomas Ravenscroft’s Melismata, London, 1611.
I remember a couple of versions of the song from school, but there are many others. This is my favourite, learned from Robin Hall & Jimmie MacGregor, which I sang around the folk clubs, schools and elsewhere in the 60’s and 70’s. It was written by Cheshire fruit farmer Leslie Haworth. I eventually discovered I was singing a shortened version, so I recorded it complete at last for the first time in 2012.