Agreed, there is no way the USA will pull out a victory. The question is only if they can pull out a status-quo peace, or how much will be lost.
Britain wouldn't take huge chunks of land. Certainly, the northern half of Maine would go to Canada. It's also likely we'll see Oregon go to Canada. OTL, Britain and America settled the Oregon Dispute in 1842. ITTL, it's likely it'll still be settled in 1842... but after a British Aroostook victory, it probably won't go the same way this time around.
Of course, this gives even more incentive for a Mexican-American War. I don't think that'll turn out
too differently, though. Given that the OTL Mexican-American war was mid 40s, only a few years after the Aroostook War would end, it'd be hard to push it much earlier. America needs time to recover. With the added incentive for California, it probably wouldn't occur too much later, either. Perhaps the US pushes for more of Baja California ITTL, though?
Unless, of course, Britain decides to rain hell on America in the peace treaty. Even with a greater need for the Californian coast, a disarmed America probably wouldn't attack a Mexico allied to Britain.
This is also going to further cement the British as America's enemy, but on the other hand it would also settle border disputes between the two nations. And the economic ties between them will be temporarily severed, but would be repaired. On the other hand, the ACW is coming up, and Britain may be more inclined to intervene in this scenario.
"Coming up" is 15-20 years in the future. A major war and especially shared defeat could also temporarily unify competing American factions, pushing the ACW even later. I think you'll see a heavier rivalry, but not too much more overt antagonism. Each side has too much to gain from being friendly. I don't see British-American relations being too different once we hit WWI (this is assuming no butterflies, of course).