Not sure, but the US would be less prosperous between 1812-1850. Britain during that period was far from free traders, which mean no free access to New Orleans, and this would hurt the US while benefiting Britain. Not to mention the Gold Rush would benefit Britain or Mexico more than the US, or the majority of the OTL oil boom would be outside the US. Many of the Irish population might decide to move to the West Coast (non US) for gold rather than the US. The US might exceed the UK (just the UK), but not Germany and the combined British Empire (much more powerful ITTL at the expense of the US). Note that a great power is still not a global power.
Of course the original US region was resource rich and would allow it to become a great power if its population exceed a certain level. If not, then it would look like OTL Canada than itself IOTL.
The scenario in which Britain control all the West means that it would control a big chunk of the OTL US gold and oil reserves. In other words, the economic warfare in the world wars would be vastly different.
The main reason why the US would be less prosperous is because the lack of easy access to New Orleans. And even then, how much would it take for Britain waive that with the right fee? It's been said that the hardest frontier for the United States to conquer was the Appalachians, and I'm in agreement with that statement, since an argument against that is rather hard.
California, here's an issue . Before the US conquered it, the non-American population was half Mexican, which of the non-Amerindians was half a mixture of various Anglos (American or British). California could persist as an independent state. This scenario is bound to make many Anglo-states in America, not necessarily aligned with the United States. They might even be British dominions. I think considering California's elite, the separation from Mexico City (whose demands were never popular on the frontier), and the Anglo influence, that California would secede from Mexico with British aid. And there were plenty of Americans operating in California before the Mexican War.
But the US will exceed Germany and the UK. The population is simply huge, the industrial capability using rivers, coal, and iron available is huge, and unlike Germany or the UK, the US has oil reserves readily available. Even confined east of the Mississippi, they will be of use. And Pennsylvania, Ohio has significant oil, and since I'd expect any state in that position to be economically advanced enough to exploit the oil in the Gulf of Mexico to some degree or another.
I think it would be a global power. It would still have a massive birth rate and a lot of natural resources. The main difference comes in that this US would probably be much more densely populated than OTL, as not as many people would move west.
Why wouldn't they move West, especially if Britain was offering them incentives?
More densely populated might be a thing, if you can get a few more million immigrants which might happen given differences in immigration laws between Louisiana, the United States, or any other state which emerges in the mix.
@metalinvader665 I am not saying it was inhabited, but that it could be. The area was mostly uninhabited except a few towns before the Louisiana purchase, at least in terms of Europeans. The north of Louisiane except on the Mississippi border has far higher than the nation's average in terms of Gallic genetics and the majority non Gallic in the north is that of African descent, so it is not necessarily something influenced by joining America.
It could, but there's still the people from Tennessee (and elsewhere) who had their sights on it and became the region's elite in time. Could Francophones from New Orleans and the region defeat these land-hungry Anglo settlers--perhaps in this Louisiana where there's an international border along the river between them. Andrew Jackson and associates could have opened up West Tennessee in the 1790s, and once a city on the Chickasaw Bluffs (like Memphis) is established, it isn't far across the Mississippi. Unless you have an international border there, and those land speculators and others would end up operating in British territory rather than American territory. Trans-Appalachia is a lot of land to fill out after all.