The Australians won't changes sides, and they won't start a war with Japan- but there'll be protests through Queensland in particular, and a lot of anger if the British government doesn't ask Japan to step away. Certainly, at the peace conference Australia will be even more pig-headed and hostile towards their Japanese allies. I'd expect that Britain and America would, at the peace conference, demand that at the very least the Japanese accept a condominium in the New Guinea mandate- but they'd probably expect Tokyo to be happy with gains in Germany's other islands that have much more strategic relevance. The Dutch and probably the French will back the Anglos.
The thing is, New Guinea was not a particular haven for resources at the time, though it certainly had potential. What it was useful for, apart from prestige, is controlling shipping routes. So Australia wasn't precisely wrong to be concerned about the territory being in the hands of potentially hostile power, though that's not to condone their hateful race based paranoia.
And Japan in this time period has no reason to want New Guinea unless it's thinking about gaining a strategic advantage over its supposed allies. So it's a recipe for tension.
So, assuming that the Japanese can take it first- and I don't think they can or will, but let's go with this- here's how the long term consequences would play out assuming the war proceeds roughly on schedule.
The Japanese are even more humiliated at Versailles. Not only are they not gaining as many of Germany's island holdings as they want, they're being made to give up a vast amount of territory. Admittedly, that's territory that's hugely underdeveloped that they have no way to make a profit from, but that won't make a difference to the blow to national pride.
Worse still, despite having provided the heavy lifting for the war in Asia, it's clear that they're valued less than the demands of white settlers. (From the Australian perspective, it's the reverse- the Japanese gave treasure and ships, but the Australians gave blood. They lost a lot of men proportionally for the mother country, and now it's only fair that the great powers recognise their sacrifice.)
The Australians will certainly not treat the Japanese with respect, just as they failed to in our timeline. PM Billy Hughes was a more direct thinker than President Wilson, but no less racist, and here he will be even more assertive in the face of what looks like a Japanese attempt to gain the ability to threaten Australia's sea lanes.
The key here is not to ask: what will be the consequences of a Japanese New Guinea in the twenties, since that's unlikely ot happen. Rather: what will be the consequences of the humiliation of the Japanese taking, and having taken from them, such a large amount of land?
This will exacerbate the political tensions both in terms of Asia's great power rivalries post war, and in Japan itself. The Anglo-Japanese alliance will have been given a blow, so expect it to fall a little more easily than in our timeline. Pan Asianism and Yellow Peril thinking will both be encouraged in their respective hothouses, though to what extent I don't know.
The clashes between the Americans and the Japanese naval strategists in the twenties will get worse- here, the Americans are clearly prepared to gang up with other regional powers to hurt Japan, and conversely the Japanese have tried to expand in a region that's not even close to their sphere of influence in a way that directly threatens the western powers.
But when it comes to seeing the long term consequences of the increased tensions... very hard to say. Everything in East Asian politics in this period ended up being contingent on teh settlement in China, and that's enormously vulnerable to butterflies.