Go day Come Day is on an Otis Spann recording, which I have somewhere. I must dig it out. It dates from 1950 or later, however.
There is no record of the song from the 19th c., either in UK or American song. The Spinners recording is the oldest I can find.

The jig doll certainly is English traditional but it must be universal in western Europe. The French brought them to Canada as early as the 17th c. French-Canadian dolls are more two-dimensional than the English dolls but the actions are similar. The name 'limber Jim' for these dolls seems fairly recent. They are also called Dutch dolls in the southeast States.
English jig dolls are very well made, and are three-dimensional as far as I know.