Joe, I think you're exactly right. The "Merrily" trail possibly leads back before 1850, and, I would bet, to England. I have looked into sources on English children's songs and games like Gomme, but without success so far.

This must be one of the earliest songs many of us heard as infants. Lucky me, my babysitter, c. 1942-4, sang me folk songs like "Billy Boy" to amuse me and herself. I have a vague memory of her singing "Merrily We Roll Along" as a separate song, with verses that basically vary the chorus like "Mulberry Bush" (which she also sang, and maybe borrowed from):

Merrily we wash our clothes / hands / face ...
Merrily we ... (can't remember any others at the moment)

-- as if it might have been one of those children's activity songs. Now did she learn that from her mother? On the other hand she was a clever lass and might just as easily have been extemporizing.

***Digressing (not fatally, I hope), I should have noted the obvious as well: rather late in "Merrily"'s life it became the lead line of the theme of Warner Bros.' Looney Tunes and Merrie Melodies cartoons. The little musical "sting," which then goes off into another melody, was still being used in 1995 for the Sylvester and Tweety Mysteries end theme.
That in turn was based on the 1937 "Merrily We Roll Along" composed by Charlie Tobias, Murray Mencher and Eddie Cantor. Not related to the Kaufman-Hart play of 1934. (Info Wikipedia)***

Back to the search into the deeper mists of time: if anyone knows of ANY use of the phrase "Merrily we roll along" (and/or its tune, apart from Mary Had a Little Lamb and sometimes London Bridge) in more songs previous to 1867 and Yale's adaptation of Christy, cite em please! "Merrily Merrily Over the Sea" and "Merrily, Merrily Bounds the Bark" are a good start. Could it be that they echo an earlier, simpler children's song or stage song with the phrase "Merrily we roll along?"   

Bob