Found it!

HE PLAYED HIS UKULELE AS THE SHIP WENT DOWN

Yo-ho, my lads, yo-ho.
I'll tell you the tale of a sailor
Who sailed the Bay of Biscay-o
As the captain of a whaler.
Of his gallant deed you all should know.
Yo-ho, my lads, yo-ho.
I'll tell the tale of the Nancy Lee,
The ship that got shipwrecked at sea.
The bravest man was Captain Brown,
'Cause he played his ukulele as the ship went down.

All the crew were in despair.
Some rushed here and the others rushed there,
But the captain sat in the captain's chair,
And he played his ukulele as the ship went down.
The pets on board were all scared stiff.
The cat miaowed and the monkey sniffed.
The old green parrot hung upside-down
Screeching "Scratch-a-Paw Polly" as the ship went down.

The owner wirelessed to the crew:
"Do the best that you can do.
She's only insured for half-a-crown,
So I'll be out of pocket if the ship goes down."
The captain answered, "Oh, you see,
It's all up with the Nancy Lee.
You're all right, but what about me?
I shall wet my ukulele as the ship goes down."

When they heard the news at the admiralty,
They wirelessed to the Nancy Lee:
"We'd like to speak to Captain Brown,
If he isn't very busy when the ship goes down."
The captain answered, "What's to do?"
The admiralty said, "At half past two,
We shall all be listening in to you
As you play your ukulele when the ship goes down."

An Irishman said, "Begorrah! I'll fight
The very first man that comes in sight."
"You can't do that," said Captain Brown.
"You have to box the compass as the ship goes down."
He shouted, "Man the rafts.
With chains and a rope you can bind them."
The bosun searched both fore and aft,
But dashed if he could find them.

At the mercy of the raging sea
Was the good ship Nancy Lee.
She struggled on against the storm.
The cold, cold wind was far from warm,
So all the crew 'cept Captain Brown
Played "Ring Ring Roses" as the ship went down.

The captain's wife was on the ship,
And he was glad she'd made the trip.
As she could swim, she might not drown,
So he tied her to the anchor as the ship went down.
Now this is the tale of the Nancy Lee,
The ship that got shipwrecked at sea;
And Captain Brown, who was in command,
Now plays his ukulele in a mermaid band.

As I said, I can't vouch for the absolute accuracy of the words, as the chap who gave me them was about 90, and remembering them from many years ago.