I don't think there is a single folk song behind this song. The first three verses quote from nursery rhymes - "Little Boy Blue", "Ride a cock horse" and "Rockabye baby" respectively; perhaps that's what gave somebody the impression it was adapted from older sources.

If I may blow (up) my own horn, here's my take on Crow on the Cradle (also featured here). The tune is my own, based as far as I can remember on a tune I set the words to when I was at school.