Here's the information I've accumulated over the years. (I would hate to have to try to find my sources for any of this.)

Jokes were not a kind of tune. There was a collection of tunes which was published, in which all of the tunes had an unusual number of bars, and all of the tunes were named Jokes of different colors. Half of the tune Yellow Joke also became a Morris Dance tune.

There were various sets of lyrics set up Black Joke. Most of them were bawdy. The clean one that my team (the Black Jokers) used to sing went:

There was an old woman, as I hear to say
She tickled the critters that came in her way
Hi ho fiddle dee dee

The chorus was the same as you have given above.

Another one that I have seen goes:

There was a woman come from France
To learn an English country dance
With her black joke and belly so white
She followed me from lane to lane
And picked my pocket quite and clean
With her black joke and belly so white

The base of her joke, it will lather like soap
And the hair of her joke will draw more than a rope
The base of her joke, it will lather like soap
And the hair of her joke will draw more than a rope
With her black joke and belly so white

I believe that I learned that version from a reproduction of an old broadside, which was titled "The Original Coal Black Joke."

I hope that's not more than you wanted to know.

Jacob

P.S. The other half of the Constant Billy verse goes, "When the fishes fly over the mountain, then will you see your Billy again."

P.P.S. Do you have the lyrics to the other Adderbury songs? The Happiest Man, The Postman's Knock, and so on?