Hi, Malcolm-
The Traditional Ballad Index covers most of the significant U.S. songbooks (and I've checked a few others), so it looks to me like this one wasn't sung in the U.S. until the 20th-century folk revival. It doesn't really fit into an American context because we haven't had that style of hunting.
The version in the Digital Tradition is a nearly-exact transcription of the text from the Weavers' Song Book (1960), with additional words and music arranged by Ronnie Gilbert, Lee Hays, Fred Hellerman, and Pete Seeger. Here are the background notes:

This one started out in the days of Robin Hood, probably, and has been sung at many a campfire since then. In the original the keeper hunted down the deer and killed her; in our version we felt sorry for the poor doe and in our last verse we arranged to let her go.

In American Favorite Ballads (1961), Pete Seeger uses the same lyrics as the first three verses in the DT, but does not include the fourth verse added by the Weavers. All that Seeger says about the song is that it is "a song from our British cousins, lots of fun as an answer-back song."
As you can see in Folkindex, all the U.S. versions are from nontraditional sources - Song Fest, Songs for Swinging Housemothers, Leisy's Songs for Pickin' and Singin', and the Weavers books and one from Florida Methodist Youth Fellowship.
It would be a real triumph to find a traditional U.S. version, but I don't think we can do it. I wish Bruce Olson were here to help.
-Joe Offer-