The British succeed in their invasion of the River Plate, seizing what would OTL become Argentina and Uruguay. Continued streams of immigrants from all over the western world over the following century eventually lead to an English-speaking majority there. Attempts to take Cordoba however fail, and a new, Spanish-speaking nation develops there instead.
Britain also takes Florida and Louisiana during the Napoleonic Wars. However, doing so creates a recurring enemy out of the Americans. Four further wars are fought against the USA in the coming century - one over the Puget Sound, which Britain wins; one over control of New Orleans, which the Americans win; one over Atlantic trade, taking place at the same time as a rebellion of American slave owners, which ends in status quo ante bellum, and a last one over control of Hawaii, which Britain wins (and where they take several Far Eastern holdings, desired mainly by Canadian whaling companies, from the Russians).
In South Africa meanwhile, the Zulus never become preeminent, thus there is no Mfecane, thus keeping a high native population instead of a depopulated countryside. This leads the local Boers to staying in place south of the Orange river. Independence is granted to most non-white areas in the 20th century, leaving whites and mixed-race peoples a narrow majority in the remaining territory. South Africa is probably the place with the worst and most acute racial politics in the whole Federation.
India was the jewel of the Empire, but the violent and protracted rebellion at the beginnig of the 20th century eventually kicked the British out. Frightened to death by the result, the Empire's elites came together and crafted a new way of organizing the far-flung Empire, giving representation to all territories in a new, federal parliament that managed foreign policy, defense, trade policy, monetary policy along with limited areas of responsibility regarding taxation and civil rights. While several areas, such as Guiana, chose independence, most decided to stick together, albeit occasionally by narrow margins.
(They also acquired Danish colonial holdings at some point, and also built a canal in Panama)
Imperial Parliament representation:
Ruled since the early 18th century by the House of Hanover (who now only enjoys ceremonial powers) and currently boasting the 5th largest population on Earth (though bound to be overtaken by Brazil, Pakistan and Nigeria within the century) and the 3rd largest GDP (behind the USA and recently surpassed by China as well), the British Imperial Federation is one of the worlds leading super-powers. Despite numerous problems and wide discrepancies, it nonetheless provides a high standard of living for its citizens and champions the causes of freedom, democracy, liberalism and free-market enterprise coupled with a strong welfare state.
So, can such a state hold together?