Actualy I doubt that. Australia is rather far away from anything and not that useful or worthwhile any trouble.
A whole continent? The prestige, even if nothing else…
basicly, if another country like France, the Netherlands or Portugal (the three most likely candidates) would colonize Australia, it is more likely that the British leave them alone or (in the case of the Napolenoic wars) return their colonies (don't forget Britain returned most of the Dutch and French colonies after the wars). Besides that, if the British want part of it, it is more likely that they just colonise another part of Australia. For example, if the Dutch founded a colony in western Australia (which was on their route to Indonesia, so it could have been useful), the British would simply colonize eastern Australia and let the Dutch be on the west.
I don't forget that. Nonetheless, I think you overestimate how moral the 19th-century UK was about such things. Even in peacetime without any legitimate excuse, even to a country that had been its ally for centuries, it was happy to threaten it with war and just take its colonies away, not because the UK lacked colonies (that certainly wasn't true) or even for the sake of any actual resources but simply for a prestige project. Even the will of a great power at least in equal in strength to the UK, namely the Kaiserreich, was treated with utter contempt by the UK in colonial affairs; the UK was happy to invade its allies (internationally recognised states) and, when it had the audacity to ask the UK to respect its economic interests there (it didn't ask the UK to not invade its allies, merely that), the UK threatened it with war and destruction until it backed down; the UK treated even France, a colonial power similar in might to the UK itself, in the same way (see Fashoda). The point is that the 19th-century UK would simply take what it wanted from virtually anyone in Europe (I say "in Europe" because the UK's treatment of the United States was rather different).