When I was searching for some original score of the music, the only bit I could find online was a short section in a quadrille called "Toodles", by Frank Musgrave. Each section was titled, implying that Musgrave simply arranged popular tunes to form the quadrille. However, it was published by Clifton's main publisher, Hopwood and Crew, which leads me to believe that it was either used by or taken from Clifton's song. The tune popularly sung is an entirely different tune, so I'm wondering whether the popular tune was fit to the song later, or whether it was the tune Clifton published in his sheet music, and the quadrille bit is just a coincidental red herring.

Below is an ABC transcription of the tune in the quadrille. You can view the full score at the Lester S. Levy Collection.

X: 1
T: Ten Minutes Too Late
T: Titled movement of the "Toodles" Quadrille
C: Frank Musgrave
S: Sheet music for "Toodles" Quadrille; Hopwood and Crew 1859
N: This is possibly the tune Harry Clifton used for his song.
M: 6/8
L: 1/8
Q: 3/8=92
K: C
(ef | gab {d'}c'bc') | (ba)a a2 a | add d(g>a) | g3-g2 g |
%
gab {d'}c'd'c' | (ba)a a2a | (gag fe>d) | c3-c2 ||