here's a rough idea.
The American Revolution does not happen; British North America stays together as one unit. The British invest heavily in North America to fortify it in the decades afterwards, as part of "Fortress North America". A parliament of sorts for most of B.N.A is tasked with overseeing this, making for less resistance than would be present if London did it themselves.
The French Revolution does not occur; the King allies with the Third Estate, most of the First Estate, and parts of the Second Estate, and crushes a noble revolt.
Another war occurs in the 1810s between France, allied with Spain (which is unhappy with Britain's actions in North America), against the UK. The British repulse all attacks on BNA and eke out a win after six years of fighting.
The war further harms the foundations of all empires in North America (including the British one); the British take Louisiana. In the 1820s, all hell breaks loose, and the Spanish colonial empire in the Americas falls. The French lose Haiti. In the early 1830s, BNA follows, with some guy bent on Manifest Destiny taking control, after leading a narrowly successful revolt against British rule (with slavery being the issue undoing British rule). By the 1840s, the choas in BNA has settled, with it all under the umbrella of some vaguely British-aligned politician. Over the following decades, the United States of North America slowly expands, with it slowly gobbling Mexico and the Carribean.