Bodleian Library Broadside Ballads has about 15 copies of this one. I don't know the tune. It doesn't mention Australia, but the convict is obviously being transported somewhere.

THE CONVICT'S CHILD

The convict ship lay near the beach.
The morn was drear and dark,
And many a wretched felon stood
Waiting to embark.
One felon stood among the rest
Whose eye was fierce and wild.
He held an infant in his arms.
'Twas, alas! his only child.

His auburn hair fell careless o'er
A brow of spotless white.
His little eyes beamed playfully
With innocent delight.
He little knew his father's heart
Was breaking while he smiled,
Or that he took a last farewell
Of him, his only child.

They tore the infant from his arms
And dragged him from the shore.
He wildly gazed around the beach
But saw his child no more.
The vessel sailed. The convict fell
In dying anguish wild.
" 'Tis done. The fatal struggle's o'er.
Alas! my only child."

The widowed mother sobbed alone [or "aloud"].
Her tears might flow in vain.
That bitter morn her husband fell
She ne'er could see again.
She pressed her infant to her heart.
Again she saw him smile.
"I'll live for that dear boy," she said.
"Alas! my only child."