The tune turns up in various campfire songs. "The Old Grey Mare", which somehow fits with the horse-a-lum, but also one I learned in he 1960s in Nova Scotia:

"I don't want to march with the infantry
Ride with the cavalry
Shoot with th'artillery
I don't want to fly over Germany,
I'm in the Queen's Navee!"

There's also a Scottish song that I found in a book (1920s or 30s) recounting the tale of Dives and Lazarus that fills out all the rhymes with "-ium". All I recall at the moment:

"To the rich man's door there came a human wreckium
Skinnamarinky doodle and a high rogeram
Wi' a hooker-doon and a grauvat round his neckium,'
Skinnamarinky doodle and a high rogeram,"

I don't think all the rhymes would have translated out of Scots, so it was probably local there.