Rob,
I have a copy of the original 18thc sheet somewhere. It has the story of how it came to be written. Off the top of my head the writer was inspired by street vendors selling those little pottery effigies of the two children which would have been based on the popularity of the older long ballad posted above. He decided to write the well-known short version at that point. If I remember correctly it is dated and if Gardiner was born in 1770 he can't have been much older than 20 when he wrote it. It came from Billy Weekes' personal collection. He very kindly did me a copy. I'm surprised he hasn't chipped in here himself. I'll look it up and give more details. It might be worth checking the tune with the Levy version which I probably have a copy of somewhere. I have lots of versions of the long ballad which was printed by just about every printer in the 18thc. Its earlier fuller title was 'The Children in the Wood, Or; The Norfolk Gentleman's Last Will and Testament, A True Story.' It was extremely popular in the period Gardiner was growing up.