Your missing the bloody point. The question is what happens to the US Civil War, not how Mexico won.
But that is an important part of it, as that war did define the latter one. By saying the British helped Mexico to such a degree as to change the war completely, we have to make several assumptions:
The British must have been absolutely they could win the quick war against an unsuspecting US, as a longer war would result in victory for the US.
The British must gain something to make up for the deterioration of the relations with one of their largest trading partners. (If not the largest; I can't remember when the threshold was crossed)
Your assumption is that the British don't fight a war in the Northwest, but the only way that the British could help the Mexicans is by fielding an army, which would be something noticed by the US government. So, by some measure, the British manage to transport thousands of troops across the entirety of the Atlantic, through the US blockade (and if the British ships attack the US ships, the ruse is off) and assemble and integrate into the Mexican army, and are able to defeat several strengthening US invasions in piecemeal.
The ruse won't hold up, and Washington will eventually find out about it. Even if the British manage to be successful enough to cause an immediate withdrawal of forces in the near term, the US still has greater numbers that can be brought to bear. And once it
is discovered that Britain is doing this, the first thing that Washington will do is prepare for all-out war.
After all, the gold rush should still happen in California, which means it is still going to be inundated with American settlers. Oregon, being more hostile and never having a line between both sides delineated, is very hazardous for US settlement (depending on
how hostile). So, the US is nearly completely cut off from the Pacific, has been stabbed in the back by their largest trading partner, and finds its citizens migrating to foreign territory in order to seek the wealth of that land.
All that will happen is, once Britain commits to the Crimean War, the US will leap at the chance to pay Britain back. And, given the choice between keeping Russia penned up, and maintaining their North American territories from the large, populous, and motivated opponent, they will choose one thing. And it will not end in Canada's, or Mexico's, favor. The US will be united by patriotic fervor for the next twenty or so years, by which slavery will slowly become obsolescent and, perhaps, be eliminated peacefully. But
that is too far in the future to judge.
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Now, I imagine that the scenario you're positing is quite achievable, if all you're wondering is what would happen to the US without the Southwest in the Civil War (which can be done without Mexico winning the war. After all, that war didn't have to be fought). The South would continue to push for the trade to be extended, especially as there aren't many territories left that would qualify. You might see the South push for a war with Spain to get Cuba (Ostend Manifesto writ large). The president and the prevailing mood would also determine quite a lot. If there isn't as much fear that the government would forcibly free the slaves, you might have some of the Upper South states stay in the union (Tennessee, North Carolina, or Arkansas) That can happen if it happens earlier or later. Really, it just depends on how desperate the slavers get, so earlier seems likely.