The US waits 5 minutes until the British diplomatics efforts disappear in the puff of smoke coming out of the barrels of the circular firing squad that was early 19th century Mexican politics. The various coup and constant anti centeralization efforts meant no government could last long enough to project any threat without the homefront collapsing around their ears, and if they tried their political rivals would use the moment (likely supported by the Americans) to impliment their own "Plan of X"
Unless Britain is willing to prop Santa Anna up like the French tried to do with Maxy, which would be a huge money sink and distraction for little concrete gain, than they don't get much. If that is the case, than they don't back down , Oregon remains contested legally while they win the settlement game, and they imposr the demand for 54 40 as they wait for Britain to blink due to needing to put out fires somewhere else in her world spanning empire.
I hear India is getting a little testy over planned reforms to use animal fat on bullets