Are asking if this could happen, or starting from this premise?
I'll take the latter...
The US is now bent on revenge and launches a massive peace time military expansion over a period of several years. Please note that the majority of the US's power and wealth remains INSIDE the US.
The usual happens. Fortifications, large numbers of advanced monitors followed by coastal battleships, machine guns arrive a bit sooner, perhaps even some early armored cars, plus a standing army worthy of the remnant US population(which is still over 20 million in 1861 and doubles very soon). Plus the occupied Americans and the Underground Railroad for other efforts.
So one fine day in the 1880s the British are having problems somewhere else in the world and the US suddenly leaps. If Canada hasn't fortified the border and introduced total conscription, the US eats them for lunch. If they HAVE, they still go down unless the British have a permanent commitment there, by which I mean something on the order of more troops than the entire British Army in OTL.
Of course, if the result of this war is a massive and permanent expansion of the British Army then the British taxpayer is liiable to see the whole situation as a disaster. We might even see an effort to appease the US before matters come to a head.
Unfortunately the minimum would probably involve so much(independent West Coast, the new Canadian territories, AND support revising the CSA border) as to be shattering to whichever British party forced the original settlement.
So the war begins. Barring massive Canadian fortification and conscription AND a major British presence(say 150K minimum), the US walks over most of Canada, retakes the territory lost to Canada and reunites 'Pacific America' in one nation. Probably most of Canada is overrun too.
Now the US offers terms. The British to recognize the reunion with 'Pacific America' and the return of the 'new territories' to the US. In addition, either some monster reparations bill or perhaps all of Canada west of Ontario to be attached to the US. If the US did really well, both of those. If matters were closer perhaps no reparations or Canadian losses but just the first items.
Now the problem for the British is clear. The US has gotten what it wants and everyone recognizes that the trend heavily favors the defense in modern warfare. The US can now assume the defensive and bleed the British white.
It goes without saying that for the British to be the victim of a sneak attack and beg for peace is absurd.
The British are not helpless, presumably their fleet is still stronger than the US fleet and can establish some kind of blockade. I'm assuming the US sends out swarms of light privateers while keeping the battlefleet, perhaps in Long Island Sound with forts, heavy guns, and mines to keep the British out?
Additionally, the CSA can be counted on for support, if only to avoid the next revenge war from a much stronger USA. If the CSA is NOT there for support, then the British really have screwed up.
The problem is simple. Once the British decide not to yield, what do they do? Is there any plausible effort by the British, even with the CSA, that will restore the status quo? Years of effort, financially crushing costs, hundreds of thousands of lives and the total collapse of London's position in the world may be considered but to what end?
A blockade of indeterminate strength and a few landings to grab a West Coast foothold or parts of Quebec won't do it. Anything less than total victory means the US won. If YOU fought a war and announced the enemy was gaining an area the size of Western Europe with the treaty, convincing anyone that YOU were the winner would be a challenge.
Let's presume naval matters go as I think, with the British unable to win a decisive victory because the US fleet refuses to be lured out. The CSA jumps in. Canada, with British help, holds a line at the St Lawrence river and most of Ontario, later retaking most of British Columbia by seaborne landing.
However, American interior lines, planning, and emphasis on machine guns and experimental armored cavalry proves a crushing advantage out west. It also becomes clear that the CSA can't afford an offensive war against superior numbers and firepower, and can barely hold out in the west after a few debacles. Possibly a major slave uprising.
Finally the Kaiser and Tsar jointly negotiate a settlement.
Under the terms, no one pays reparations(but the US seizes British property inside the US and regained territories, Canada keeps current boundaries, plus Washington State. The Dakotas, Montana, Idaho and Wyoming revert to the US. 'Pacific America' returns to the US. Colorado, Utah, Arizona, Nebraska, Kansas and Missouri are yielded by the CSA, along with portions of Delaware, Maryland, and West Virginia.
By OTL standards this US would be much smaller and weaker in potential. By comparison with what prevailed for 20-30 years such concessions could only be seen as a shattering defeat by the British and CSA without even having the benefit of less tension with the US.