The (presumably) traditional tune was obviously adapted (from a "traditional" tune - also used for the song "Mr. mcGuire, Sit Down")by Edward Harrigan's father-in-law David Braham, a prolific composer of music for popular songs and musical conductor at Harrigan's theatre. Several other of Harrigans songs have adapted "traditional" melodies. The original "Regular Army O" was performed as a song/sketch. The sketch portion is long lost but it would seem to have been sung by some of the male members of Harrigan's Theatre Comique troup - each performer taking a verse in turn and all joining in on the chorus. Attached is a link to a picture of the songs original performers James Bradley, Edward Harrigan, Tony Hart and two others whose names escape me at the moment. The print is an exact copy of the original photograph which I have somewhere.

https://www.flickr.com/photos/harrigan73/1231514117/