I'm pretty certain you're correct, Nigel. I could easily check. However I do believe the French connection is a lot older and is not so surprising, Cornwall's connections with Brittany.

Dunstan's Cornish Songbook pt 2, p43, but only a few of the carols in there are claimed as native Cornish.

Jean Ritchie sang a version from Kentucky. It is possible this derives from the Cornish version.