It's called "The Old Dun Cow Caught Fire" by Harry Champion

Some mates and I in a public house
Were playing dominoes last night
When all of a sudden in the pot-man came
With his face all chalky white
"What's up?" says Brown. "Have you seen a ghost?"
"Have you seen your Aunt Mariah?"
"Me Aunt Mariah be blown," said he.
"The bloomin' pub's on fire!"

Oh there was Brown, upside down
Knocking back the whiskey on the floor
"Booze, booze," the firemen cried
As they came knocking at the door
Oh don't let 'em in till it's all mopped up
Somebody shouted "MacIntyre!"
And we all got blue blind paralytic drunk
When the Old Dun Cow caught fire

Old Johnson rushed to the port wine tub
And gave it just a few hard knocks
He started taking off his pantaloons
Likewise his shoes and socks
"Hold on," said Tibbs, "If you want to wash your feet
There's a tub of old ale here
Don't wash your feet in the port wine tub
When we've still got some old stale beer"

Oh there was Brown, upside down
Knocking back the whiskey on the floor
"Booze, booze," the firemen cried
As they came knocking at the door
Oh don't let 'em in till it's all mopped up
Somebody shouted "MacIntyre!"
And we all got blue blind paralytic drunk
When the Old Dun Cow caught fire

instrumental

Just then there came such an awful crash
Half the bloomin' roof gave way
We were doused with a fireman's hose
But still we were all gay.
So we got some sacks, and some old tin tacks
And we bunged ourselves inside
And we all got drinking good old Scotch
'Til we was bleary-eyed

Oh there was Brown, upside down
Knocking back the whiskey on the floor
"Booze, booze," the firemen cried
As they came knocking at the door
Oh don't let 'em in till it's all mopped up
Somebody shouted "MacIntyre!"
And we all got blue blind paralytic drunk
When the Old Dun Cow caught fire

Fire! x8 during instrumental

And we all got blue blind paralytic drunk
When the Old Dun Cow caught fire.

Hope that helps. I don't think it is a trench song though.