Here's an attempt, and I stress the word attempt. It probably has holes in it big enough to drive a tank thru:
1600s: Australia is discovered early in the century. Many European nations lay claim to it, but no one really tries to enforce their claims. With the conclusion of the 30 Year's War, a surviving Gustavus Adolphus forges an alliance with Cromwell's England. (We have seen in other works
that Gustavus is perhaps capable of getting along with those opposed to the monarchy.) The alliance is one of mutual support; Gustavus needs help in solidifying his hold in northern Germany and against Russia; Cromwell needs 'anti-Papist' allies on the continent to offset France and Spain.
The so-called 'North Sea Alliance' authors the Pacific Accord - a treaty guaranteeing the fair division of Australia and all other Pacific discoveries. France, Spain, and even the Dutch all sign the Accord, hoping to placate each other with what they consider a meaningless piece of paper. After all, who wants a few little islands and a continent-sized desert?
1700s: Cromwellian Britain doesn't do as well against France here. There is something similar to the 7 Year's War, which the French emerge from mostly victorious, especially in India). Butterflies give the American revolutionaries (which rebel over religious differences) a little more oomph in their victory. New Brunswick, Nova Scotia, and Newfoundland become US states, while Labrador is absorbed by Quebec. Without as much British domination in Canada, more and more Tories flee to South Africa and South America. With only token trading colonies in Australia as of 1800, most of the British 'undesirables' who would have been sent there OTL instead migrate to Africa and Patagonia (where the Brits are trying to create a new dominion).
1800s: British Republican propaganda and French bankruptcy after the ARW cause the overthrow of the French monarcy and the rise of a 'Napoleon' figure, who embroils Europe in a couple decades of war. By 1820, 2 major European forces arise: a coalition of Britain, Sweden, and Russia; and a French 'empire' consisting of France, Italy, Switzerland, Austria, and lower Germany.
Desperate for funds, the Brit coalition gives the US gov't 'ownership' of Australia in direct contravention of the Pacific Accord, in return for $$$. Unwilling to risk another devastating war, the French do nothing.
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I don't have the energy to go any farther, but I forsee a much more rapid growth of the 'US' in this TL. They have not just one, but two continents to practice a manifest destiny on. By 1850, the 'US' could conceivably control North America (maybe they buy the rest of Canada from an ailing Cromwellian republic) solidly, and Australia and NZ on paper, with settlers on the way...
How's that?