The changes needed to domesticate bison are not that unreasonable: they're the same genus as cows, so it's not unimaginable to say that they're domesticable. They need to be less ornery, but they also need to a. live in mixed-gender herds b. be perhaps a little more hierarchical and c. they need to be less prone to randomly deciding to run at top speed for miles at a time (which would make taming them for early farmers nigh impossible without horses).
I don't think the Mexican deserts need to be much of a barrier for them. Environments do change over time, and at one point the deserts were green enough to allow a massive southward migration of Nahua people. Domesticated bison brought to the southwest could be bred to be more desert-friendly, and good herders taking good care of them could bring them across the desert. I do agree that the valley of Mexico is likely to be their most extreme southward outpost of their presence, though.
The time that bison would be domesticated would depend on whether they are more like horses or like cattle for the people who domesticate them. According to my hobbyist farmer uncle it is possible to ride them, and they are fast and agile, so they may become riding beasts as much as meat and milk beasts. If they are cattle-like, they can be reasonably domesticated after 2,500 BC (when farming appeared in North America), if like horses, an earlier domestication is possible, somewhere between 5,000 and 2,500 BC would be reasonable.
As for zoonotics, Buffalo provide a number of nasty bacterial zoonoses including Lyme disease, TB, and Brucellosis. Buffalo transport could create the trade routes necessary to spread these diseases and keep them alive. I have not been able to find any literature on viral zoonotics that they pass to humans (though believe me, I looked), though them having a pox virus, parapox virus, or measles variant is not impossible. Although it hasn't happened OTL, the genetic changes that would make them domesticable in an alternate universe might cause butterflies that let them catch and carry a viral disease into North America, which they then pass on to humans.