What a nightmare Keenir!
If Ben Chifley (PM 1945-49) had had the popularity to swing a republic there would have been no stopping him. He’d have nationalised the Banks (his attempt to do so in OTL was his downfall) and an Australia of massive engineering projects, centralised plans and broad socialist policies would have been the result.
Between ‘49 and ‘66 you can forget it unless you find a way not to have Bob “I did but see her passing by…” Menzies as Prime Minister, which is pretty much impossible.
Harold “all the way with LBJ” Holt may have gone for a referendum but wasn’t in long and was busy with the ladies.
Blackjack McEwen only had five days so that’s unlikely.
John Gorton, nope. Popular enough to possibly swing things that way if he’d lent support to such a movement but not inclined.
Billy McMahon, even less likely. (Only Australian Prime Minister to die “on the job” with the Mrs if the rumours are true.)
Which brings us to Saint Gough the divine. His fans will tell you he was deposed in a CIA plot precisely so he couldn’t make Australia a Republic, withdraw from Vietnam, end poverty and introduce the Age of Aquarius. In reality he initially became PM with a majority of eight seats, and that was further reduced to five seats in his second term. He was then dismissed and an election called that resulted in the largest swing against the Labor Party in post war history.
But you face a suspension of disbelief issue with this.
Since a change to the Australian Constitution needs a majority of voters in a majority of States and a majority overall to pass it is unlikely in the best of circumstances, under a divisive or controversial Prime Minister it’s impossible.
Menzies, one of Australia’s most trusted and politicly strong Prime Ministers tried to ban the Australian Communist Party with a referendum and was defeated. And that was at the height of the Cold War!
It is just really, really hard to convince Australians to change the old Constitution unless they can clearly see something isn’t working. Especially if they suspect the change of giving more power to Politicians in general, and Canberra Politicians in particular.
My best bet for an early Australian Republic would be in the 1920’s during the boom when Stanley Melbourne Bruce was at the height of his popularity.