Might be hard pressed to find anybody (over a certain age, anyway) who didn't sing a form of that song at school! The best-known would still be the set that appeared in Baring-Gould and Sharp's English Folk-Songs for Schools in 1905. It has been much re-printed, in all manner of songbooks, since then.

The set you refer to does look as if it's had part of an American version of The Carrion Crow grafted into it. Some years back, an extract heard from a Midlands group was posted at Daddy Fox; it's the same, and like as not they got it from the Wayfarers. I'd be surprised if this were a traditional variant.