The only "Aussie Version" of Rudolph I've ever heard was the one we used to sing at carolling time or school concerts (or both) up in the bush where I grew up.

It went something like:

Rudolph the Red-nosed Reindeer       (Reindeer)
Had a very shiny nose                (Like a light globe)
And if you ever saw it                (Saw it)
(You would even say it glows          (Like a light globe)
All of the other Reindeer             (Reindeer)
Used to laugh and call him names      (Like Pinocchio)
They never let poor Rudolph;          (Rudoplph)
Join in any Reindeer Games            (Like Monopoly)

Then one foggy Christmas Eve
Santa came top say                   (Ho, ho, ho)
Rudolph, with your nose so bright: Won't you guide my sleigh tonight

Then all the other Reindeer          (Reindeer)
Laughed and shouted out with glee    (Yipp - ee!)
Rudolph the red-nosed Reindeer       (Reindeer)
You'll go down in his-tor-y          (Like Captain Cook)

All the bits in parentheses are "echoes" to the verse lines and are spoken (or called out en masse) with energy - not sung. Almost like each little 'addition' was a sudden inspiration or 'bright idea'

Muttley