For the origin of the Copper's "The Old Dun Cow caught Fire", you nust go back to Harry Champion. But the words and music are attributed to Harry Wincott and there are two versions, one like the Coppers and one like 'Arry.

Oh, there was Brown, upside down
Mopping up the whisky on the floor
"Booze, booze, booze" the firemen cried
As they come a knockin' at the door
"Don't let em in till it's all mopped up"
Someone said to MacIntyre.
And we all got blue blind paralytic drunk
When the Old Dun Cow caught fire

MacIntyre was presumably nearest the door. This is roughly the same as Harry Champion.

And there was Brown upside down
Lappin'' up the whiskey on the floor.
"Booze, booze!" The firemen cried
As they came knockin' on the door (clap clap)
Oh don't let 'em in till it's all drunk up
And somebody shouted MacIntyre! MACINTYRE!
And we all got blue-blind paralytic drunk
When the Old Dun Cow caught fire.

The Coppers, more or less'

Harry Champion's tune is rather different to the Coppers and, of course, is sung faster.