The "in the van" reference puzzles me - it is not in any version I know, except the "Bushwhackers'" one. I have occasionally heard that rendered "in the Band" - but that does make much more sense.
The original collected words appear to be those Alan Scott collected from Mrs Susan Colley of Bathurst in 1956 and published in Singabout, volume 5, #4, p.13 as Woolloomooloo (and which are in my anthology; Singabout, Selected Reprints, ed Bob Bolton, Bush Music Club, Sydney, 1985).
I happened to be born on a very frosty morn
Quite contagious in the town of Woolloomooloo.
And it was in Riley Street where the folks first heard me bleat
For at the time I'd nothing else to do.
My name is Dan McCartey I came from the Old Darty
My father drives a carty when he's any work to do
But he is very lazy mostly drunk and nearly crazy
Gone wrong along with the boozing throng down there in Woolloomooloo.
Every time that he'd get tight then mother and him would fight
Half their time they used to spend in jail,
They were known to the police for they always broke the peace
And not a soul would ever go their bail.
When I grew up a lad I went straight into the bad
Soon became a most accomplished thief
The government was kind and didn't seem to mind
For in Darlinghurst they granted me relief.
I was watched with constant care and they used to cut my hair
For six months I wasn't allowed to roam
But my visits I'll renew 'twixt there and Woolloomooloo,
And in either place I'll find a welcome home.
The tune is the usual variant of Killaloe
The words that constitute the chorus in the version posted by Alan are in fact a separate item collected by Ron Edwards in 1957 from a Mr Brisbane - in Melbourne! It seems that this Sydney song was only sung by those who were not in Sydney ...
The text seems to be a performers' text that grew organically from these 2 collected texts.