The following song was mentioned by Flewruby above as the source for the tune of "I Don't Want to Join the Army":

These lyrics, without a title, come from Gender and Power in Britain, 1640-1990 by Susan Kingsley Kent (London: Routledge, 1999), page 275:

On Sunday I walk out with a soldier.
On Monday I'm taken by a tar.
On Tuesday I'm out
With a baby Boy Scout;
On Wednesday, with a Hussar.

On Thursday I gang oot with a Kiltie.
On Friday, the captain of the crew.
But on Saturday I'm willing,
If you'll only take a shilling,
To make a man of any one of you.

A couple of other books say the title is "On Sunday I Walk Out with a Soldier" and that it was sung by Gwendolen Brogden in the Palace Theatre revue "The Passing Show" in 1914.

Different lyrics, and a different title, are found in The Ones Who Have to Pay: The Soldiers-Poets of Victoria BC in the Great War 1914-1918 by Robert Ratcliffe Taylor (Victoria, BC: Trafford Publishing, 2013), page 132:

TO MAKE A MAN OUT OF YOU

On Sunday I walk out with a soldier.
On Monday a sailor for a pard.
On Tuesday of course
With a B.C. Horse;
On Wednesday, a Home Guard.

On Thursday I gang oot wi' a Kiltie.
On Friday, a Fusilier or two.
But as you've all been willing,
It didn't need a shilling,
To make a man of every one of you.