I just happened by your thread as I was searching for any way to identify (composer, year, melody, etc) a song my mom sang as a grade school child in Northern Saskatchewan 1938-1940. Unfortunately, this cobbler song doesn't seem to bear enough similarity to any of the above variations to be considered a re-arrangement or anything else. Frustrating, as I am typing the lyrics to 7 songs so far, from my mother's childhood handwriting; and to this point, the only one I've been able to identify is "Some Folks Do" by Stephen Foster 1855. Here is what I have.

The Busy Cobbler

VERSE 1
As he hammer all day long
The busy cobbler sings a song
All goes playing through and through
To make the shoes as good as new

CHORUS
Ratatat, Ratatat,
Tap tap tap tap tap tap
Ratatat, Ratatat,
Tap tap tap tap

VERSE 2
At his bench, the busy cobbler
Makes the bits of leather fly
Though the shoes seem past all mending
Still, he sings as all go by

*REPEAT CHORUS
Ratatat, Ratatat,
Tap tap tap tap tap tap
Ratatat, Ratatat,
Tap tap tap tap