Yep. Environment is the key
Australia would be in a marginally better position in terms of sustainable carrying capacity minus rabbits and other introduced vermin. Whack the people who introduced rabbits, foxes, etc. over the head with a clue wand and Australia would have more potential. Ultimately though, water is the key issue, followed by poor soil and a fragile environment that has only recently had to cope with animals from the rest of the world.
I suppose some kind of early breakthrough on cheap bulk desalinization could help with the water issue, though getting the water where it was needed would be difficult and capital-intensive. Some kind of change in climate patterns that left Australia well-watered? If that change left North America and Europe arid, that could do it. I'm not sure how feasible that kind of climate change would be though, or what would cause it. Much of Australia was once covered by tropical rainforest, but that was at a time when it was considerable further south and North and South America were separate. I'm not sure which of those elements were crucial, but changing either of them would undoubtedly lead to an unrecognizable world in terms of human political development.
Just tossing ideas around here: I suppose you could have the US get nukes early (late 1930s) and use them to blast a new sea-level Panama canal. I'm not sure that (a) that would be feasible, (b) it would allow enough ocean current between the Pacific and Atlantic to make a difference in climate, or (c) That it wouldn't trigger some kind of nuclear winter type scenario.
In any case, somebody having nukes before World War II started has huge implications in terms of the course of that war, and is unlikely to lead to a world anything like our current one. World War II as a nuclear war could lead to Australia being one of the few unscathed areas, which could make it a major power relative to the devastated countries that took the brunt of any nuke exchange. The unscathed part would be tough to manage though unless the politics of the war changed considerable.
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Dale Cozort's Alternate History Newsletters - nine years of Alternate History ideas, scenarios, and fiction.