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Had we stayed a colony of Britain, what would have been the chances of a Louisiana Purchase and the immigration from Europe?
Unlike the Americans the British wouldn't have needed to buy Louisiana from the French, they would have conquered it and included it as an add-on to their empire at the Congress of Vienna.

No successful American Revolution, no French Revolution, no Napoleon, No Congress of Vienna, no European power vacumm, Purssia remains another small Germanic kingdom, No German Empire, No Triple Alliance to counter it, No First World War, No Bolshevik Revolution, No Hitler, No Second World War. Basically it's the situation in the Turtledove/Dreyfuss The Two Georges

It's a world of Empires, the heresy of republicanism never gets a look in. Withour that troublemaker James Monroe spreading dissention south, Spain retains South America.

The British Empire acquires the Ottoman Empire and the Chinese Empire as Protectorates to contain the Russian Empire

The Industrial Revoluition takes place in England on schedule in the 19th Century, but blends into the 20th one in the North American Union dominion, making the British Empire the world's industrial superpower.

No successful American Revolution, no French Revolution,...

That does not follow right there. The French Monarchy and French society had their problems, which culminated in the French Revolution. How the French people would have attempted to solve those problems in the absence of a successful American Revolution is unknown, of course. I mention the Butterfly Effect. But revolution was not something invented by the American colonials and completely unknown before them.

My comment about the British conquering Louisiana was facetious, of course. Again the Butterfly Effect. No one can predict with any certainty the events that would have followed a successful British suppression of the colonials' revolt.