With any POD you want, have the United States control territory in the British Isles, or European mainland.
PoD: Edward of Kent dies without offspring, and William of Clarence predeceases George IV. Ernest Augustus of Cumberland succeeds to the throne as Ernest I.
Cumberland was a hated reactionary. Britain at the time was in a state of ferment, with many people prepared for open rebellion if parliamentary reform didn't pass. (OTL it passed by one vote.)
With Cumberland's baleful influence against it, it fails. Violent disorders break out. King Ernest, backed by the Duke of Wellington, rallies sufficient military and police power to suppress the disorders.
Britain decays into an authoritarian reactionary police state. Vast numbers of liberal Britons emigrate to America, included many who are wanted by the political police.
The U.S. becomes the champion of democracy and freedom in the "anglosphere" against the repressive British Empire. Ernest's Britain responds to the 1837 rebellions in Canada with savage repression that alienates Canadians; a second rebellion with U.S. support succeeds, and Canada joins the U.S.
When the Great Famine strikes Ireland, the British government flatly refuses to relax the Corn Laws and allow tariff-free grain imports. Starving Irish become indentured workers in English factories. Any who get out of line are dumped in Australia as convict labor. Many flee to America, where the sufferings of Ireland become legend. Some Irish are dumped in New Brunswick and Nova Scotia; this leads to those colonies joining the Second Canadian Revolution.
The Whig Party is more successful than OTL; instead of Harrison dying and Clay losing by a hair, the Whigs have two solid Presidencies and become the dominant party. The U.S. still acquires Texas, Utah, and California as settlers in those regions throw off Mexican rule and apply to the U.S.
But it does not lead to a sectional crisis. There are four new free states in former Canada, and immigration of British dissidents has shifted political discourse to the left - against slavery. When Missouri, Maryland, and Delaware adopt gradual emancipation laws, some Deep South Fire-Eaters call for secession, but get little support.
Some of them form a conspiracy with British agents, as the British government wants to cripple the U.S., secure its cotton supply, and regards the U.S. as nest of subversion. But the plot leaks out, further poisoning relations and discrediting secessionism.
Kentucky adopts emancipation. West Virginia separates from Virginia and adopts emancipation. The new western states of California, Oregon, and Columbia are free, though Texas is slave. There are 28 free states to 11 slave states. Slavery is squeezed out gradually.
By this time, Ernest is long dead, but Britain is locked into the political culture he created.
Australia is a prison state, with a handful of planters ruling over a mass of convict and indentured laborers. When the revolution occurs, circa 1890, the U.S. first recognizes the rebels and then admits Australia to the Union. The great distance is a problem, but the telegraph, steamship, and railroad make it possible. (Less travel time than from Washington to New Orleans in 1812.) A Constitutional Amendment provides for representation in the Senate only and a local Congress for local matters.
The British government is very angry, but can't do much about it. Britain is not as much of an industrial giant as OTL.
Sometime in the 1900s, Britain merges with rising Germany and does become an industrial giant, but still as repressive as ever. Imported German police prop up the regime. Overt slavery in the West Indies has been abolished, but white supremacy and de facto slavery remain.
The U.S., having shed overt slavery, sees a civil rights movement start a generation later, and succeed rather quickly - in part because of Britain's bad example and the ongoing influx of liberal immigrants.
Friction with the U.S. over Britain's brutal repression in the West Indies leads to open war. The war ends with the U.S. liberating Britain, which joins the U.S.