Why does Texas remain independent? That's the likely point of divergence. Once this is determined, you can start to extrapolate the next steps.
Sam Houston was a strong proponent of annexation, as were the many Southern Democrats who rushed into the Texas Republic to claim free and near-free land. But Houston faced strong anti-annexation winds from the Whigs in the US. They were against bringing in another slave state.
But what if Sam Houston is too badly wounded/killed at San Jacinto, or what if David Crockett misses his appointment with destiny at the Alamo? There were others, like Mirabeau Lamar, who dreamed of a southwestern Texian empire. What if Stephen F Austin had lived, or Lorenzo de Zavala? Several key political figures died in 1836 or 1837 and if they had lived, their vision would have been different than Houston's.
But back to the tin, the most likely reason for Texas to remain independent is for the Whigs to retain the presidency between 1842-1848. Of course, that would have serious repercussions on the US. If the Whigs were successful at blocking Texas Annexation, it would, I think force Texas to court good relationships with Britain and France.
One myth that gets trotted out is that Mexico would come crushing back in and re-annex Texas. Assuming immigration patterns similar or better than OTL, Texas's irregular forces were able to defeat the Mexican army of the era at battles like
Battle of Salado Creek and
Battle of Hondo River. Also, by 1845, Texas' population was around 3 times larger than it was in 1836.
Source. The point is that every year Texas grew stronger. Mexico's problem with invading Texas is the same that existed in 1836. They were at the end of a very long and very poor supply chain. Sure, Mexico could recruit tens of thousands of men to serve in the Mexican army, but until the advent of the Railroad or the ability to control the Gulf of Mexico, they lacked the capacity and logistics to field that army outside of central Mexico.
Of course, Texas suffered from a similar problem. Their failure to capture Santa Fe was tragic but expected, given Texas' poor preparation. The Mier expedition also showed Texas really didn't have the ability to project power outside of its population centers.
My point is that if annexation is off the table, Mexico between 1836 and 1860 is poorly positioned to pose more of a threat than they did in 1841 and 1842 IOTL. @
Everdarklegion, if you're interested in developing a timeline, figure out who the president of Texas will be. Then figure out if it is possible for the politics in the US to veer away from the pro-annexation side. Once you do those things, then you can connect the dots to the future.
If you're interested in how I did it (aside from the ASB handwavium that sets up the story), you can see my timeline here:
Forget the Alamo - An SI ISOT