Any conclusion, even IanC's, may be altered by further evidence, but in practical terms the burden of proof is on the challenger. It would be safest to say, "On the present evidence, it appears to descend from the broadside ballad of 1837-45," but the caveat of new evidence is often merely assumed. Without specific evidence, we shouldn't just assume the song is older (regardless of generalizations like, "You can't date folklore"; when you get as specific as this, sometimes you can).

So the appearance of the song in the 1967 film, "The Adventures of Moll Flanders," set in the mid 18th Century, is another Hollywood anachronism - as is the playing of "Blow the Man Down" on the concertina in the same movie.