I've read that the earliest surviving published version of the song is dated 1700. The tune, as noted above, goes back a little earlier. "The Bath" appears in the 1651 edition of Playford and is replaced by "The New Bath" in 1689. The latter is a different dance, with a different tune.

Quite a lot of tunes had currency across much of Europe anyway, so who knows where it originated. Certainly some unknown 20th Century buskers in Lille aren't likely to give a good provenance for a tune that's been around over 200 years. I was in North Italy last year and a hurdy-gurdy player in the mountains was playing what sounded like an English tune. Turned out it was. He got it (directly) from the hurdy gurdy player in Blowzabella.

:-)