UK v US up to 1880
The figures are dollars to dollars, not dollars to pounds.
Amazing how the "US is split in two and has lost a war" is the default in this sort of question...despite the track record of failure of British arms in the Americas in the Eighteenth and Nineteenth centuries, as the wars of 1775-83, 1806-07, and 1812-15 make clear.
Cripes, they couldn't even beat the Argentines...
Best,
Well, the UK never fought an all-out war against the US. The War of 1812 was in part a sideshow to the Napoleonic Wars. TBF, the army defeated at New Orleans did contain many of Wellington's best regiments so it does indicate the problems Britain would face in trying to defeat the uS. The American War of Independence saw Britain face France and Spain, and it placed a greater priority on defending the Caribbean sugar islands than the war in BNA. The Buenos Aires episode is quite funny in a sense, definitely not the finest hour of the Empire!
BUT, the period from 1830-1860 or so sees the British Empire at the height of its power relative to other countries. Not a global hegemon, like the US attempts to be today, or a military superpower. But the 'workshop of the world' with the largest navy and the main financial centre.
So IF Britain deployed all its power in North America it should be able to defeat the US, Not conquer or occupy it permanently, but force it into making peace. The Royal Navy could blockade the Atlantic and Gulf Coast after seizing isolated ports or islands - much like the Anaconda Plan of Winfield Scot (sic) for the Unionists in the ACW. It could deploy sufficient troops to defend Canada and make punitive raids into US territory etc.
However, this would require a massive effort and I can't see why it should make it without extreme provocation from the US. At that time Britain was more interested in opening new markets for Free Trade and consolidating its control over India. Which was the secondary core of the imperial system and the one form which a lot of expansion was generated. As far as BNA was concerned, Britain would be happy simply to trade with the US and invest in it. So it would be fairly conciliatory of US interests as shown over Oregon. Of course, IF the US was to try to conquer Canada I can see Britain reluctantly going to war. Whether even then it would put in the effort needed to "win" I'm not sure.
After the 1860s I think the US becomes so powerful that it conquer Canada and ignore British retaliation. Militarily at least, economically maybe less so.
I'm open to correction on that end-date, would the US have been strong enough to actually defeat Britain before then rather than just be too big to conquer?