Here's the tune in ABC, as best as I can do. I hope it makes sense to people who read ABC. It strikes me as a graphic argument for the superiority of music written out on a staff.

N: Bridgework
M: 4/4
K: A major
L: 1/8
C: unknown
S: Elliot Forbes, who died recently. He conducted the Harvard Glee Club & Radcliffe Choral Society for years, and he and his wife used to sing this as their party piece. I asked him for a copy, and he sent me one - but didn't include composer or lyricist.
O: unknown, probably a novelty song from the 1930's
N: 1) This is in A major, and I am NOT indicating F#, C#, and G#, as they are included in the key signature.
2) I don't know how to notate tied notes, or dotted notes, so I'll use +, e.g.:
F/2+2E+E/2 = a 16th note on F [actually F#, of course] tied to a quarter note plus a sixteenth note on E
3) Bridgework is not quite the same as dentures. It is a false tooth, or a group of false teeth, which attach to the remaining teeth at either side of the gap where the missing teeth once were. And, of course, one person's bridgework won't fit another person! Even if the other person were missing the same teeth, everyone's mouth and dentition is unique. So, in the unlikely event that a cowboy possessed bridgework, his mother couldn't use it.

Verse:
E|EEEF CA,2E|FFA+A/2F/2 F/2+E2+E/2E/2F/2|FFA+A/2F/2 FE2C/2C/2|CB,B,A, B,3E/2E/2|
EEEF CA,2E|FFA+A/2F/2 F/2+E2+E/2E/2/F/2|FFA+A/2F/2 FE2C/2C/2|CB,B,A, B,4|
Chorus:
C2zD EcBA|BB7|d3d cBAB|c8|
C2zD EcBA|BB7|d3d cBAB|A8|
B+B/2A/2B4A2|F/2E/2+E7|B+B/2A/2B4c2|e8|
C2zD EcBA|BB7|d3d cBe+e/2c/2|A7