To create an exact timeline of an independent Texas, you'd need to establish an exact POD. I think the best one would be that David G. Burnet, with the full support of then President Lamar, was elected Texan President in 1841 over Sam Houston. Houston was in favor of annexation, Burnet wasn't. This would cut U.S. annexation plans off from the beginning.
To have a future, Texas would have to resolve its issues with Mexico; most likely by war. Here's how I see it. At some point from 1841-1844 (I think this has to happen before the 1844 U.S. Presidential election) Texas and Mexico go to war. Texas troops go on the offensive, with most of the battle taking place in the states of Coahuila, Nuevo Leon, Tamaulipas. Even heavily outnumbered, untrained, and undisciplined, Texan militia fight A LOT better than Mexican troops (1/4 of which will desert before any action takes place). Besides, the Texans had better leadership, and most were veteran troops from the revolution; plus, the Texan military had fought the Mexicans, and were knowledgeable in city fighting (something the Americans didn't have when they invaded). So after defeating the Mexican Army a few times in the northern provinces, political chaos in Mexico City causes an overthrow of the government, and the new government (probably led by the military) immediately wants peace on Texan terms. I'm going on this assumption because in OTL, Santa Anna had to retreat to Mexico City to prevent a military coup during the Mexican-American War. And in any long-term war, Texas doesn't stand a chance, no matter what.
In the peace treaty, Texas gains undisputed control of its claims, and I think it would annex the top row of Mexican states and everything above them (maybe not the Baja peninsula). They would also probably stick Mexico with their enormous debt, so we'd probably see the French Intervention happen a few years earlier.
This would end any talk of American annexation, and it would cement Texas' hold on the continent (no matter how weak it is). MAYBE there would be a new constitution to take in the fact that Texas would now be a multi-state republic, but this is open to debate. I do think that Congress would divide the newly acquired northern territory into future states. California would probably decide to remain part of Texas, as long as most of the inhabitants are given proper rights as a fellow state in the Republic.
In the 1844 election, with Texas annexation no longer an option, Tyler doesn't do anything. With expansion no longer the main subject, it would probably devolve into the Whig's economic policy vs. the Democrat's. Polk wouldn't be nominated, and the odds are that Van Buren would be. The Whigs would still nominate Clay. I think Clay would be elected, just because of Van Buren's record with the economy. This would permanently upset American history. There's no Mexican-American War, so there's no new territory; there's no new territory, so the debate over the legality of slavery in the territories never happens; with this debate gone, the Whigs won't split over the issue and they'll remain the main opposition to the Democrats; so no Republican Party, no Civil War, no war heroes, nothing. This leaves the slate of American politics completely open to Alternate History speculation.
I can see Texas either being pro-British or pro-German. Britain would be their natural ally, and it would benefit them economically. But I think like in OTL, we'd see A LOT of German immigration into Texas, maybe even more in this TL. And if the French intervention happens earlier like I stated above, I don't think the Texans would enjoy having a new Mexican Empire propped up by the French on their southern border; I highly doubt there would be a war or anything like that, but there would definitely be a lot of antagonism there. That would send them on the side of the Germans, who were as anti-French as could be. But all of these possible alliances would develop in the future.
As for race relations, that can go a lot of ways too. In Texas, there would be three main races: Whites, Blacks, and Mexicans. So there would be a race on top (the Whites), one on the bottom, and one in the middle. The one on the bottom wouldn't be able to vote, would get the terrible jobs, and for a long time they would either be slaves or a cheap workforce with no rights, depending on how you go. The ones in the middle, would see some better treatment. Not as good as whites, but they'd probably have most rights, they could testify in court, maybe vote, etc. But Whites will be seen as superior socially no matter what (so no intermarriage or anything like it). So it just depends on whose on the bottom: the Blacks or the Mexicans. Texan Whites, now possessing a large amount of territory filled mostly with Mexicans who have no allegiance to the government, may turn to the Blacks to balance things out. With the same language and both groups being Protestant, the Whites may see this as the best way to counterbalance the Catholic Mexicans. Likewise, the White Texans, having spent several decades living among Mexicans and being heavily influenced by their culture, may decide to keep the blacks enslaved and give the new Mexican residents more rights. But I highly doubt that both groups would be held at the bottom by the Whites; its one or the other. Like some of the other things, this is up for debate.
So what does everybody think?