Who’s the man with the big red nose?

The most enigmatic of the songs collected in F.T. Nettleingham’s Tommy’s Tunes (1917) is, in its entirety, this:

(Click the picture for a better view)

Wondering what this was about, I’ve searched the Internet, and found an Australian drinking song – I think the sort where you have to down the pint before they end the verse:

Who’s that man with the big red nose?!!
Oo Ah Oo Ah Ah!!
The more he drinks the redder it goes!!
Oo Ah Oo Ah Ah!!
Down down down down,
Down down down down……

I’ve also discovered that Irish Boy Scouts used to sing this (Maybe they still do?)

Who’s that man with the big red nose
Who ha who ha ha
The more he rubs it the more it grows
Who ha who ha ha
There ain’t no flies on us
Oh no
There ain’t no Flies on us
Oh No
There may be flies on some of you guys but there ain’t no flies on us
Oh NO!

I’d like to know more about the song. Was it specifically a WWI song, or was it one of those that came from the old Regular Army? Or was it a civvie song imported into the War?

But most of all, I’d like to know if this little chant was in the mind of the great borrower, T.S. Eliot, when he wrote Sweeney Agonistes:

You dreamt you waked up at seven o’clock and it’s foggy and it’s damp and it’s dawn and it’s dark
And you wait for a knock and the turning of a lock for you know the hangman’s waiting for you.
And perhaps you’re alive
And perhaps you’re dead
Hoo ha ha
Hoo ha ha
HOO
HOO
HOO
KNOCK KNOCK KNOCK
KNOCK KNOCK KNOCK
KNOCK
KNOCK
KNOCK

Well, the new edition of Eliot’s poetry will be published soon, with voluminous annotations by Christopher Ricks and Jim McCue. I wonder if the man with the big red nose will be in there.