My father was a WWII vet, and he used to sing this. I was always meaning to get the whole thing from him, but I never did.

I could swear that the President came into it somewhere. Did Old King Cole call for the Pres-i-dent in the final verse? Or did one of the topmost ranks make some comment about the President? "Get me the President" or "Ring me the President" would scan. But I may be inventing that. I don't find it in anyone else's version.

I don't remember what rank said what, beyond the first few, but it definitely had:

The army's gone to hell, said . . . (but it wasn't chaplains; I don't think there were chaplains in my dad's version)
Where's my boots and spurs, said . . .
We do all the work, said . . .
I want ten(?) days leave, said . . .
Right by squads, squads right, said . . .
Hut-two, hut-two, hut, said . . .

and of course:

Beer, beer, beer, said the privates
Merry men are we
There's none so fair as can compare
With the fighting infantry.