My fellow Mollyhawk, Sue Griffiths, came across your enquiry, and passed it on to me. She knew I could tell you about "Timmy", because I wrote it.
I can't claim credit for the basic idea - that came from Vinny, of Landlocked. Nor can I claim the tune - it's loosely based on a version of "The Plains of Waterloo" In fact, I can hardly claim credit for the words, either, given that they consist of a series of hoary old jokes crudely spliced together. But apart from all that, the song is entirely mine!
The words are as follows - please feel free to do what you will with the song.

TIMMY
(David Robertson)

We sailed away from Barking creek, this old sea dog and me,
The roughest, toughest mongrel you could ever wish to see,
Not what you'd call a handsome dog, but one of noble fame,
Where'er you sail, the shantyman will always know his name.

Chorus
Timmy! Get down, boy!
Timmy! Get down, leave it!
Timmy was a dog of noble fame,
Timmy! Get down, boy!
Timmy! Get down, leave it!
Every shantyman will know his name.

While Tim was chewing on a bone, the bo'sun's mood was black,
He was barking and a-growling - and the dog was growling back!
"Oh make him drop it! cried the bo'sun, Call him off, I beg!"
For the bone that he was chewing on was in the bo'sun's leg.

In all his natural functions, Timmy took a special pride.
He'd never cock his leg but that was on the leeward side,
And when he'd squat, he'd pick his spot, on sloops and square-sail traders,
And you couldn't walk the poop-deck without pulling on your waders.

One sunny day, on deck he lay, a-licking of his tackle.
The mate looked on with envious eyes, which raised old Timmy's hackles.
"If I could do that, cried the mate, I'd die a happy man!"
Says I, "Give me a fiver, and I'll ask him if you can!"

The mate on board a Yankee barque once gave old Tim a bone,
He gnawed upon it half an hour, then wandered off alone,
To bury it was his intent, the daftest dog afloat,
For he's dug right through the bilges, and he's sunk the bloody boat!