I don't think the US would go to war with both the UK and Mexico. They'll probably postpone the Mexican war to fight the UK though I don't know how succesful that'd be.
I agree. Can't see the US being mad enough to fight both powers at the same time.
In the unlikely event they did as pointed out in comments the key blow was the landing at Vera Cruz and advance on Mexico city. Say goodbye to such a force if at war with Britain. Blockaging the entire US coasts would be difficult but US trade would be seriously crippled and if they got ashore before Britain knew about the attack VC would definitely be blockaged. Coupled with supplies to the Mexicans and the knowledge of a powerful ally I think even if the US managed to storm Mexico city they would face huge gulleria resistance, steadily sapping their strength.
Far more likely would be a was between Britain and the US only. If the US seek to occupy all of Oregon do they also attack in the east? If they don't then Britain can reinforce Canada easily. If they do then they give up any chance of avoiding a big war.
Given the British links in the disputed area, established colonies there, interests of the HBC and also the importance of a Pacific coast for the Canadian colonies, not to mention that Britain couldn't afford to allow a naked land grab I would see Britain fighting even if the Americans try for a limited attack.
As Dave H and others have said the US would be in for a hell of a lot of grief in a war with Britain. Barring some dramatic pre-war build-up, which would cause a lot of discontent I suspect, the minimal US forces would have no capacity to seriously threaten eastern Canada. Large numbers of militia could be quickly but they have a history of being highly unreliable, especially for operating outside their own states let alone in invasions of other countries. Also once Britain starts imposing a blockage and possibly raids on military targets I can see the southern states especially, unhappy with the war anyway, being very unwilling to allow their militia to operate in attacks on Canada. Similarly probably many of the norther coastal states will want to keep troops back to protect their homes and be angry at the devasation of their trade. Only the internal northern states may be in favour of the conflict and they will have pretty limited resources.
Once Britain ships forces across the Atlantic they will be able to drive the US out of any gains in the east. This is before the Crimean conflict so probably a fair amount of incompertence exposed but lessons will be learnt and the country has too many resources. Raids into the US may suffer losses from determined defences but are likely to also do a lot of damage and errode willingness to continue the conflict.
In the west its a no-contest. If the US move troops west and attack 1st they may make some gains but its virtually impossible to reinforce or supply them overland, or by sea once a British force starts operating in the region. In India there are tens of thousands of unemployed troops from the various wars there, many with experience of fighting under british control, and also the recent experience of shipping such units to China for the Opium war. Western Canada is a lot further but still withing capacity to expect to see the US forces driven from all of Oregon either into the wilderness further east or forced to surrender.
As such I expect Britain to take all of Oregon as a result of the war. It may also possibly establish a base at hawaii as a staging point. Possibly annex more of the Indian territories, say pushing the border east of the Rockies to the 45th parallel or take small regions in the east to secure Canada against future attacks. Depends on how angry the British government is, how distracted it might be by events elsewhere and how long before the US cries uncle.
You might get a war with Mexio later which the US would probably win as it would have gained important experience from the conflict. Especially since with Oregon cut off that's the only way it would get a Pacific coastline. However there might also be some deterrants to such a development. After a costly war which has left sizeable amounts of the economy in ruins the appitite for such a conflict may well be reduced. Furthermore, since this would be seen as giving gains mainly to the south - wrongly as it turned out in OTL - it might be even more unpopular in the northern states. Also, given that British eyes have been drawn to the US might Britain intervene to stop the US winning such a victory. [Even the chance of such an intervention would make a landing at VC a very risky affair so the US might have to advance overland, a much more difficult process].
The other question is if the defeat hightens sectional interest - south objecting to the war and north to the south's lack of support might you see an internal crisis earlier?
All in all such a scenario would be bad for the US but good for Britain. Longer term effects depend so much on butterflies.
Steve