AIUI, I think the idea is that, VA being on the fence of secession for a period OTL, combined with the fact that Taylor is from Virginia, would lead to unionists having a decisive advantage.
They didn't really hesitate once the shooting started, which (if Taylor can be taken at his word) would be pretty quickly in this TL.
Also, Va was a firmly Democratic state, and the Dems tended to be the secessionist party.
One other change. Kentucky is another strongly Whig state, so may well be pro-Union from day one, rather than faffing about with "neutrality" as in 1861. So Taylor has a chance to intervene in support of Tennessee Unionists, a chance that was missed in OTL's Civil War. Tenessee's Civil War history may be more akin to Missouri's, with an attempt at secession which fails.
All this, of course, assumes that the South goes ahead with secession. However, I'm by no means sure that they were ready for that yet. You may just get a southern version of the Hartford Convention, a lot of talk but nothing much done. In that case, the Territorial question pretty much goes by default, and New Mexico may become a Free State some timwe in the 1850s.
One interesting point. Does Utah also get early statehood? Mormon Polygamy was little known about in the east (we didn't "go public" about it until 1852) so a Utah admitted then probably has no laws against it. Things could get lively once the pols in Washington learn what manner of state they have admitted.