As far as I can tell this song grew from a dance, sometimes identified from Danish as Shoemaker Boy. It's centuries old, and may be classified as one of those occupational dances which I guess were around then according to Iona Opie.

Wind Wind Sugar Baby is how it was sung to me over seventy years ago in New York, In the cradle. It was sung medium tempo. slower than the dance. minus the second part.

I'd love to know more how and when the Danish migration brought this to english speaking shores.

Apparently to bring clarity to this ostensibly cobbler's dance, the melody of "Bobbin up" dropped a note which "Sugar baby" retained. The gestures are mostly preserved in both.   

The Danes eventually stuck lyrics to the tune and as far as I can tell translate something like "first it's this-a-way then it's that-a-way..ugh ugh..." and then there's a clapping or tapping or both

You can find youtube examples of much of this... none, naturally, are exactly the same... iI'd love to hear anyone's personal experiences with this wonderful old piece that still bobbing and winding and surviving.

By the way I've taught it as a two part round, with the gestures... fun.