I realize I may be making a gross assumption, but my feeling has been that the TD Rice "Jim Crow" connection was so likely (if not "obvious," for those familiar with it) that it could go almost without saying.

That being said, while I believe the "Jim Crow" is a trope that originates with Daddy Rice, its occurrence in this specific song doesn't clinch any particular origin. At this point, I think English speaking singers could have slapped on "Jim Crow" (i.e. the floating phrase of minstrelsy) as a way of rationalizing incomprehensible or nonsense Swedish. Just as the nonsense in the Swedish versions could have been a way of dealing with the unfamiliar "jim crow."

Given the Swedish (/Danish, accord. to Carpenter) and German connections, I wouldn't be surprised if there is a bit more that can be said about the song after exploring those channels.

Sternvall (the Swedish chanty collection editor) attributed one version to 1875.

On the level of text -- that is, ignoring the issue of what was historically and culturally likely -- *I* think it may be easiest to imagine the song starting off as a a Scandinavian or German one.