Obligatory reminder that the British colonisation of Australia was not purely as a dumping ground for convicts. It was also established to be a naval base to project power into the South Pacific and to resupply vessels in the area. That reason still exists even if no convicts are ever sent there, as does the sense of competition with France which meant that Britain planted a colony on any part of Australia where the French showed an interest. (And also in New Zealand, for the same reason). The odds are high, though not absolute, that Britain will plant some sort of colony anyway, though the nature of that colony would be different.
In terms of other players, the Dutch knew where Australia was since the early 1600s, and never showed any interest in colonising. I think they can be safely ruled out. The French were exploring the South Pacific repeatedly during that period (Bouganville, Dufresne, La Pérouse, Baudin, etc), and made some claims, though how seriously they would have followed up on those claims I have no idea - though the British certainly believed they were serious and tried to stop them. Germany wasn't seriously interested until much later.
