The Point of Divergence for the United States of Australia timeline can be pinpointed as 3rd December 1854. In our timeline, that was the day of the Eureka Stockade battle, where police stormed a stockade formed by mine workers as part of a protest against the exorbitant fees they were forced to pay, plus a hefty dose of racism thrown in directed against Chinese workers.
In the real timeline, the battle ended with the miners either being killed or scarpering away, but the carnage was fierce. It also left a marked legacy as one of our first national trials-by-fire (we weren't a nation at the time, of course). Over the years, Eureka has come to be regarded favourably and even warmly by Australians, who see it as the first battle for freedom.
In this timeline, it was something a bit more spectacular. In this timeline, the Eureka Stockade began what became known as the Eureka Rebellion. The Rebellion literally forged a nation.
In this timeline, Peter Lalor, the miners' leader, is killed in the Battle of Ballarat on December 3, 1854. Lalor, along with Henry Ross (who died at Eureka in OTL as well) is considered as a martyr, and his death only galvanises anti-government sentiment among miners. Before long, an armed revolt is going on across Victoria, and it soon spreads to other states. Leadership of the miners falls to someone else (I haven't determined who yet), who becomes Australia's George Washington, leading the Reform Army, later the Australian Continental Army against the British. The miners are inspired by the American Revolution and its ideals, and adopt U.S.-style institutions as they fight for independence from Britain.
Spurred by Irish nationalism (the Irish were numerous in colonial Australia, especially in Victoria) and later by simple patriotism and the "spirit of Eureka", the rebels successfully manage to secede from the British Empire. The civil war in Australia, or the "War of Independence" or "Eureka War" goes on until 1859, when the Treaty of Ballarat is signed, establishing Australia as independent, though it only consists of the Eastern states - WA and SA (including the NT) don't join the new nation right away. The revolutionaries, still led by our Down Under Washington, establish a United States-style governmental system, with some uniquely Aussie tweaks, and the United States of Australia is born. Over time, an Australian equivilant of Manifest Destony takes hold, and Australia expands to cover the whole continent and areas of the Pacific, including PNG, Fiji and New Zealand.