I agree with those who think that the British will attempt to claim the whole continent later if not sooner, and that the runners up for grabbing some part of it and in British default, the whole thing, France, are distant second runners, since they rarely succeeded in getting large numbers of French settlers to emigrate to colonies under French control. I do not know if the number of French immigrants to the USA, or other independent countries or other country's colonies were proportionate to their settlers in their own colonies, or not. But the only French colony I know of that got really large numbers of Europeans was Algeria, where they remained outnumbered by native Algerians--and where in order to pump up the numbers of Europeans, the French Third Republic invited other Europeans, with mainly southern Europeans responding, to settle there too. And this discrepancy in colonial emigration was evident in the low numbers of settlers in New France and Louisiana before the demographic drop in French birthrates in the 19th century, so that phenomenon merely underscored a deeper, longer trend.
Other nations in Europe, notably the Germanies and Italian states, were sources of much emigration, but lacked an established colonial system; their emigrants would settle, as I left open the possibility a large number of French emigrants may have, other nations' colonies or in independent nations like the USA or the Latin American nations. If Britain unaccountably left Australia alone until very late in the 19th century, acting as a dog in the manger only to deter French claims, or claims by other old established European colonial powers in the region (the Dutch, the Spanish, the Portuguese) I suppose that sometime after Bismarck's downfall the Germans might take an interest, or perhaps earlier, Italy, but the sensible thing to expect the British to do is to cement their claim by having some settlements there long before any other powers but France have any strong presence in the colonial game that far afield. Even settling at a tenth the OTL rate, that should place enough loyalist bases there to deter shoestring operations by other powers.
OTOH, I also agree that if Georgia and other North American territories remain open for penal colonies of Britain, they will probably not divert them all the way to the Antipodes and thus the character of Australia as a set of British colonies would be quite different, more like New Zealand only without the detente with the native peoples.
But--we might need to look more closely at how and why Britain is able to hang on to the North American colonies. I believe it is going to be very hard to avoid some kind of showdown with American secessionists eventually, because the problems at issue were not trivial, and simple decrees from Westminster that the colonials were expected to simply obey would not resolve them well. I fear it would take something like an attempted American secession to get the British ruling classes thinking seriously and effectively about what it would take systematically to keep those colonies under control, and the outcome would be timelines where Britain had more resolve to win, perhaps at the expense of losing other assets they held or expanded OTL. And that having defeated the rebels, they would work out rather harsh means of staying in control, involving maintaining heavy forces there paid for by taxes and requisitions extorted by those same forces, in league with an invidiously maintained local elite amounting to a kind of aristocracy, based on loyalism during the Rebellion. I also think an element of control would be exercised by alliance with Native tribes, who would hit demographic bottom (due to diseases) and then increase in number while being culturally assimilated to British society--as a privileged collective aristocracy themselves.
The upshot is that dumping cumulatively one quarter of Britain's entire massive 19th century emigration in the form of presumably disgruntled convict laborers into this stew might seem like a risky thing to do, because the convicts might join forces with disgruntled American colonial subject masses and overwhelm the classes set above them. Even if British leaders share confidence that such rebellions can in the end be put down, bringing in the dry tinder for them in the form of penal colonists might seem not only risky but also a breach of faith with the various entities set in place to keep order in America. Thus, perhaps somewhat later than 1788, but probably not a lot later, certainly by the time of the Napoleonic Wars at the latest when other reasons to set up shop in Australia were also providing motives, the drive to use Australia as a penal colony will pick up. I suppose America will continue to get some convicts, but the bulk of them might well wind up being diverted to the Antipodes after all. In this case, Australia's history will differ in detail but not in essence and the overall outcome there will be quite similar.
Except perhaps that without the successful American Revolution to uphold and exemplify republican ideals, and with my belief that on the contrary the manner of governing America will become distinctly less democratic, and more strongly oligarchic, perhaps in none of Britain's colonies, no matter how distant or how European-like its demography becomes, will there be democratic local autonomy. The British ruling system may well take the lesson from the American rebellion to be that a firm hand must always be in place, even if costs more money, and so there would be no place for autonomous Dominions under the Crown--colonial regimes will be legally under strong Crown control, and practically an alliance of the interests of the central powers in London and the handpicked recognized colonial elites, who I think will probably remain canny enough to reinforce their strength with pragmatic rule, ensuring a good business environment winning the regime the support of middle classes without necessarily giving even them, let alone the ruled working class majority, significant political privilege--that would be reserved for the most successful and demonstrably loyal. No democracy in Australia, Canada, South Africa or the OTL USA--Canada will be in a continuum with the other American colonies of course; privileged perhaps as regions most of whose populace stayed effectively loyal and quiet. Or at any rate very little democracy, and that subject to being overridden or dissolved should things seem to be getting unruly.
Note I'm not necessarily talking about a dystopia, exactly. I think retarding Anglophone democracy (and I also expect a firm hand in the colonies will blow back to Britain, retarding advances of the masses into the voting classes there too) is a bad thing in and of itself, and that it will be accomplished by rather terroristic means. But there may well be compensations of various kinds, perhaps. Anyway it would be quite a different society than OTL I think, and the world as a whole, with the British example reversed on these points, will be a lot less liberal politically across the board. Perhaps the French, in defiance, will figure out how to manage to maintain a genuine universal manhood suffrage Republic earlier, or at any rate much the same as OTL (genuine broad based parliamentarian republic after 1871 or so).