Proud? That goes against my understanding of Irish culture. I though that almost all Irish people hated the British then.
Nope, that's the romanticised myth put about since 1916. There was an extremely strong nationalist movement, which included a vocal anti-British minority, but the majority of nationalists wanted an Ireland that remained within the Empire.
Remember, it was the British Empire not the English Empire. You'd find Scots, Welsh and Irish all across the globe- and not just as soldiers, sailors and labourers. One of the things that gets lost in the Kelly mythos was that a huge proportion of the colonial constabularly was Irish, and often Catholic- go and look up the surname of the policeman the gang killled.
The real break-down in relations didn't come about until the aftermath of the rising. Even the fall of Parnell didn't destroy the pro-monarchist sentiment.
Also, remember- at this time, Irish people in the rest of the empire were more likely to be anti-republican than the Irish at home were. If you were an Irish-Australian or an Irish-Canadian who was feeling discriminated against because people suspected the Irish weren't loyal, you wanted to prove that just because you were Irish that didn't mean you couldn't be British or Australian or what have you as well.
Don't get me wrong, the exceptions are good fun to talk about- the Hokitika funerals in NZ, for example- but they are exceptions.
I envisage the relationship between a Republican Australia and the UK as being not dissimilar to the one in OTL. There'd be a more of an emphasis on gestures of independence- if a war breaks out, then expect actual deliberations in the Australian parliament (or equivalent) before Australia joins the UK's side. No Menzies as-Britain-is-at-war-Australia-is-at-war-also. But Australia would still join the war.
If the Royal Navy wanted to base a squadron in Freemantle, than this Australia would expect a formal request to be made. And then it would be promptly granted.
There's a movement for Australian independence, but not really one for Australian seperatism, if you follow.
As for including PNG but not NZ- that's unlikely. The British kept slapping down Queensland's attempts to get Papua because they saw that it was being driven to a large extent by anti-German paranoia, and that it would require British shipping and investment to make possible. An independent Australia might make the attempt to get Papua, but I doubt they have the resources- expect it to be even more undeveloped. On the other hand, the plight of the pacific islanders in the plantations might get far worse which is an unpleasant thought.
Moreover, Australia doesn't have the population to colonise in force. If you read newspapers of the time, one of the most obvious things you'll take away is the huge paranoia about the lack of population. People thought that the Russians or the Chinese or the Mongols (yes, really) were going to come and wipe out the empty cities.
If there's enough surplus population in, I don't know, Brisbane to go and colonise somewhere than it's far more likely that they'll just be given land grants in the interior.
This brings us to the question of the relationship with the Aboriginals- expect it to be just as bad, if not worse. Those contemporary Australians in favour of the republic were mainly of the type catered to by
The Bulletin, whose slogan was "Australia for the White Man." They believed that the Aboriginals were all going to die out by halfway through the twentieth century, and the most that could be done was to ensure that their passing was not too painful.*
NZ is far more likely to be an integral part of Australia than Papua, but it's difficult to say what will happen. As someone upthread said, it all depends on Australia. If Australia's been driven to Republicanism by some major POD in Britain, that might affect NZ. If it's all about who Victoria does or doesn't send to the Federation conferences, little might change.
Hope there's something useful in all of this- my main point is that there's no one flash point that would take Australia to being a republic- you'd need something that would affect the various colonial societies on a systemic level.
*Our most brilliant writers and thinkers of the ninteenth century, ladies and gentlemen. How depressing is that?