Well, here's a Website with John Dengate lyrics. I don't see one called The Melbourne, but maybe one of these others is about the ship.

And here's another list of Dengate lyrics - no Melbourne.

This video (click) has two Dengate songs about the Melbourne.

-Joe-

A start at the first song:

What does the Melbourne on a cruise off Jervis Bay?
She sails on the briny blue with the Voyager in the way.
So, it's hard a-port (for who that's aboard???)
On a peaceful summer's night.
The destroyer would sail or a carrier failed
To give way on the right.

Oh, the weather was fair for a bosun's chair,
So the admiral went for a ride.
He swiped (?) all hands to elastic bands
That loomed on the starboard side.

The ship ... tis the enemy
[no idea]
So they cut her in half, just for a laugh,
And drowned a third of the crew.

Pipe the compass, port the (?)
All that nautical stuff.
The whistle shrieked and the captain (leaped?)
To the bridge in a (???) huff

And West by west (?) is the course that's best
So, come on, all you men
There was great distress in the officers' mess
That night in the R.A.N.