The bits from jeffp and Open Mike
You remind me of a man.
What man?
The man with the power.
What power?
The power of hoodoo.
Hoodoo?
You do.
Do what?
Remind me of a man.
What man?
etc.

and

That's Life.
What's Life?
A magazine.
How much does it cost?
Twenty-five cents.
I've only got a dime.
(shrug) That's Life.
What's Life?
etc.

were also heard around Melbourne in the 50s except that the magazine one didn't use dismal guernsey; as I recall, the price was
"Three bob"
and the response was
"I've only got a deena."

And the antiphon/response of
"See ya later, alligator.
After a while, crocodile."
above was always
"See ya later, alligator.
In a while, crocodile."
although the added response of
"Not so soon, you big baboon."
did appear at some stage in the 60s.

Ian Turner's book of children's playground songs "Cinderella dressed in yella" (1969? and out of print now) has a huge number of such sayings (including most of the ones mentioned above) but has no reference to any of the versions of Puddin' Tane. The only time I ever heard of "poon" in Oz was among a particular group of students in Canberra in the early-mid 60s. "He's a real poon!" was used in the same sense as "He's a real dill!" I have no idea of how or why it appeared although JFK's presidency, well perused in Oz, may have had some connection.

Cheers, Rowan