Actually I'd say the USA is screwed. Big time.
Britain is distracted at this point, with the Sepoy war, the beginning of the Irish famine, and the 1848 revolutions. But it still has the capacity to blockade the Atlantic coast in the USA. Heavily affecting New England. It also has a larger fleet and army in the Great Lakes. The US army at this point is rather small and will mostly be made of volunteers, takes some time to gather those up. BNA is much better equipped and prepared for an early strike.
While the US will likely have advantage in Oregon. Britain certainly has the advantage in the Great Lakes / Atlantic Coast. The longer the war drags the war is worse for the USA.
You are also forgetting this would happen at the same time that any conflict with Mexico would be happening. Even in the US manages to delay this conflict some crazy Mexican leader is bound to try and invade Texas at some point.
This is in the mid-late 1840's?
The HEIC is engaged in the final conquest of the Punjab, but this requires no additional crown forces. A war would be a boon to the Irish as it would open the recruiting offices and cause the embodiment of the Militia, so money would flow to allow the purchase of food. The Canadians still haven't fully stood down their army from the 1837-41 affairs and are much stronger comparatively than many other projected US-UK wars in this period.
US military potential in this scenario is a known, because they did mobilise around this period.
The regular army has 12 infantry battalions, 1st-8th US Infantry and 1st-4th US Artillery. They have 16 field guns (all 6 pdrs) manned by the light companies of the 4 artillery regiments. The mounted force consists of 2 regiments of Dragoons and a newly raised regiment of Mounted Rifles, none of which is trained for mounted action but all of which are infantry with horses.
Congress authorised 8 new "one year" infantry regiments (9th-16th US), a rifle regiment and another Dragoon regiment.
The states contributed usually a volunteer regt or two on average. I'd expect they'd do better in event of a shooting war with the UK, but much of this will be absorbed by the need to man their coastal fortifications.
Oregon is lost. They are effectively isolated from the US for military purposes and the British can occupy whatever they want. The Californian Republic may get their way and be annexed to the UK rather than US.