Joe caught the error in the Traditional Ballad Index: the date of the Minerva Smythe entry is 1852, not 1812.

As noted above, the earliest date I have found for sheet music is 1847, E. Ferret & Co., Philadelphia. Since the composers are English, it is probable that the first printing was a little earlier, but I have not found an earlier English printing (1848 Australian sheet music noted above).

The "answer" posted by Joe was titled Cheer Up My Own Jeannette in sheet music published by Benteen of Philadelphia (n. d.), copy at Levy Sheet Music (link in first post).