The end of the American Revolutionary War was finalized by the signing of the Treaty of Versailles in 1783. This divided up the land in the New World, setting the boundaries of the various nations therein:
- England lost its colonies, though it gained/retained a few islands.
- France was left with a few minor gains, but not much in the new world.
- Spain regained West Florida; their land stretched from modern Florida to most of mid-to-western North America, down through Central America into South America.
- Finally, the United States gained more than it expected, thanks to the award of the Northwest Territory.
From that point on, Britain would not be much of an influence, nor would the Dutch. The French retained what would become Canada, but lost Louisiana to the Spanish. Spain not only regained West Florida, but French Louisiana as well.
The entire gulf coast of North America, from Florida to Texas and far to the south, belonged to Spain. In 1800, France secretly purchased Louisiana from Spain - an area stretching from modern Louisiana to Montana. However, without a "united wallet," the individual states would never have been able to afford the purchase; France would have kept their territory. In the short term, France would have lost money on the deal, but in the long term they would get a strong stream of income.
Without the incursions into Florida by the US, Spain would keep control of that area: Spanish Florida. Escaped slaves and native refugees would make up a majority of the population, along with Spanish immigrants. Without interference, the country of Florida would eventually be more fully settled and defended by Spain. Divided by France's Louisiana, it wouldn't have taken part in the 1821 Mexican War of Independence.
Texas would still become a Republic in 1836, breaking away from Mexican control. However, since the weaker Independent States wouldn't then annex it, the Mexican-American war would never take place, and thus Mexico would continue to hold California. With the gold rush just a few years later, Mexico would become much more powerful, retaining and building up the land between our Mexico and California.
The Northwestern Territory would eventually become several states, though with much different borders; it may even become one large country. With plenty of land, water, wood, and natural materials, the area would grow quickly.
As the northern states moved from farming to factories, they would have little use for slaves, though southern states like Georgia would probably continue to own slaves. The American Civil War would never happen. If any slave revolts occur, the slaves will run to Florida, joining the indigenous population.
French Louisiana (as well as much of the country to the north) would continue growing, bringing in money for France. As trade expanded up and down the Mississippi, the two French nations would eventually merge together. With land from south to north, France would be making much more money, with much greater span of income generation. Farming, furs, hunting, timber, and taxes on any goods going through the Mississippi river would make a lot of money, all funneling towards France.
War of 1812
Rather than between the USA and England, the war of 1812 would be between the French Colonials and England. With French pressure from Europe and a huge base and revenue stream from the North America French, England would lose terribly. In our world at that time, England controlled much of the northern-most of North America, including Rupert's Land; in this world, the war of 1812 would kick England off of North America entirely. That would leave the northern land to the French territories to envelope the entirety of what we know as Canada. The Independent States would be sheltered from conflict, though probably offer France aid, as France took on all comers: France vs. Spain, France vs. Britain, even France vs. Russia as it expanded west. Towards the middle of the century, Canada/Louisiana would merge, and shortly (and successfully) declare independence.
Spanish Florida would gain independence as well; its people would be a mixture of native Americans, Africans, a few French, and the Spanish. It would be a much wilder place - no retirement homes and snowbirds there! I imagine it as a blend of modern Mexico and Brasil; a few major, modern cities along the coast, mostly populated with Spanish and French, and many tiny villages near the swamps and forests, populated by native Americans and Africans. If any conflict occurs, I imagine the French (now Canadians) will shave off a bit of the westernmost side.
The Independent States would most likely remain independent, but as their population grows, their borders would not. Cities like New York City would never happen; the state of New York isn't large enough to support such a massive city on its own. The nations would need a lot of trade to survive, and would form a strong coalition.
Additionally, the northern French tended to make peace and trade with the natives, rather than drive them off or enslave them; while they would still have trouble with the more nomadic tribes, the Trail of Tears would probably never occur.
By 1900, the land from California to Mexico would be a rich, growing nation. Corruption would grow just as quickly, and without the US to support the 1910 Mexican Revolution, the country would be a much different place. The rich northern California would become the new capital; the south's metropolitan population would shrink as the rich moved to the nicer, cooler north.
As King-Ink mentioned, the State of Deseret would probably succeed; it may not be as large as originally intended, though. My guess is that it would cover what was the Utah Territory. As a border country between Canada and Mexico, it would actually have a greater chance of survival. Neither would really want the desert there, and it would act as a buffer between the two.
Canada would be rich as well - richer, in fact, stretching from Louisiana to Canada along the Mississippi, from Oregon to Michigan. A SuperCanada, if you will. The border with SuperMexico from California to what is today's Oklahoma would be tense, more so than the border between the US and Mexico was, since both countries would be much richer and much larger. Mexico's land stopped at the northern end of California; our states Oregon and Washington would be absorbed by Canada. The Oregon Trail would be much hazier; many immigrants would start in Louisiana, the largest port, or St. Andrew's, the closest Canadian port to the east. Still others would drift from the Independent States, passing through to reach Oregon.
World War One
The first World War would be radically different. Mexico would likely be a much earlier ally of Germany, and would probably attack SuperCanada. While the USA wasn't be around to get them angry by meddling in their affairs, SuperCanada was. With the greater strength of a larger, much richer nation, Mexico would be a force to be reckoned with.
In Europe, the more powerful France and Spain would be able to turn back Germany earlier, but in North America, the war between the two superpowers would be rough. The Independent States and the Republic of Texas would take up arms against Mexico, but wouldn't be able to help much in the war in Europe. Florida would likely stay out of any fight.
In the end, the superior factory power of the States and SuperCanada, and the oil from Texas and SuperCanada, would crush Mexico. Without as many natural resources as the combined American Allies, Mexico would lose a lot of land; Texas would take the gulf side of Mexico, and SuperCanada would extend south from Oregon into California. Mexico would be left with the (poorer) southern half of California, Arizona, New Mexico, and the western half of modern Mexico. Its greatest resources would be gone.
The Great Depression would not occur; though the droughts would be harsh, the new resources from Mexico would push SuperCanada to greater heights. Post-war SuperCanada, Texas, and Independent States would flourish. Without the destabilizing effect of the Great Depression, World War II would not occur until much later. As more oil is discovered along the Mississippi and North America, SuperCanada quickly grows into the top superpower, with resources to spare. The Republic of Texas hits its oil boom. Mexico, already poor from the earlier war, would eventually split into several nations.
Hawaii, meanwhile, will likely remain its own nation, a small, poor country in the middle of the ocean. While Russia would try to claim Alaska, SuperCanada would have better access. Alaska would be very similar to how it is now - used for oil and military bases, but otherwise largely left to the natives.
World War Two (1950s)
By the 1950, the biggest threat is not Germany, but Russia. Without World War II to kill off their population, Russia grows much more powerful, eventually trying to expand into Ukraine and SuperCanada, and sparking this world's second World War. The sides are much different; for one thing, Germany and Russia have switched sides. Instead of fighting Fascism, the Allies are fighting Communism. German engineers create the atomic bomb, but it never needs to be used; Russia's European and Asian allies are much weaker than Germany's allies, and the war is over before the bomb is complete.
North America would be vastly different; in fact, the whole world is a much different place. The Americas, Europe, Asia, even Africa have different borders, even different countries; France, having made it through both wars relatively unscathed, is a much stronger nation. Germany, having rebounded from its failure of WW I and joined with Europe against Russia, isn't the "bad guy" any more. England would be slightly weaker, having lost its stake in the New World much more quickly. Russia would be destitute, and probably would become several small, poor nations. With the major differences in Europe and Asia, politics would take a drastically different turn. SuperCanada would be a superpower, if not the superpower; the Republic of Texas would be a rich nation, rolling in oil. With the close relationship between the North American countries, they would all (save Mexico) be quite well off. The smaller nations would likely form a conglomeration similar to the European Union. Florida may eventually turn itself around; being next to the richer nations may make it a resort destination, in which case it will eventually become similar to the Bahamas. If it doesn't embrace tourism, however, it will remain poor. If Deseret manages to hold on long enough, it will eventually become a stable nation, though resource-poor.
The language of North America and the world would be very different as well; SuperCanada would speak French almost exclusively. The Republic of Texas would speak both Spanish and English. The Independent States would mostly speak English, though the northern and western states would speak some French, and Georgia would probably speak some Spanish. Florida would speak Spanish and English, as well as several odd dialects. Mexico would speak Spanish, for the most part.