Using a POD of 1787 instead of 1836, we can reasonably assume that the United States could have been stopped at the Mississippi had Napoleon been unwilling to sell Louisiana in 1803. With Spain remaining a "stong" presence in North America until the end of the 19th century, Spanish Florida could have remained a Spanish possession well after 1819. Without the addition of the Louisiana territory, the United States probably would not have made a claim to Oregon, allowing it to remain a British possession. Texas could possibly have become an independent nation if the same original Anglo settlers arrived as had in the 1830's. With the discovery of gold in California, Americans, Louisianans, Spaniards, and the British would all have flocked across the continent for the gold rush. The mixture of nationalities in California would conflict, but the Bear Republic of California could have become an independent nation during the 1850's. The Confederacy might not have formed in 1861, however, due to the lack of turmoil in the states and territories west of the Mississippi River - no Missouri Compromise or Bloody Kansas. However, the Mormons in Illinois would still have been ostracized, and ended up in Utah as they did in OTL. The nation of Deseret would most likely have been allowed to exist by the Mexican authorities, since the Mexicans had settled the area very sparsely. With the defeat of Napoleon in Europe, he could have fled to Louisiana in 1815 and established himself as the Emperor of Louisiana, rather than been exiled to Elba. If this were the case, and his nephew became Emperor later on as in OTL, the merging of the two Empires would have been possible. The installation of a French puppet as the Emperor of Mexico would have led to the formation of a large French Empire stretching from the Canadian border down to Central America, dotted with a few smaller republics such as Texas, California, and Deseret. Of course, without an United States stretching from sea to sea, Seward would not have purchased Alaska, resulting in the presence of a Russian Alaska as well.
Therefore, by 1870, North America will have a varied landscape of political entities:
United States of America - from Maine south along the coast to Georgia and inland to the Mississippi River.
Empire of France and Mexico - from the Gulf of Mexico to the Canadian border, west to Deseret, Oregon, and California, south to include Mexico.
Republic of Texas - occupying the state of Texas.
Republic of Deseret - occupying the state of Utah.
Bear Republic of California - occupying the state of California.
British Canada - all of Canada, including Washington and Oregon.
Spanish Florida - occupying the state of Florida.
Russian Alaska - occupying the state of Alaska.