hmmm... if the US lost a war to the UK and didn't get the OR territory, and also didn't get the western territories from Mexico, then the US will be a middling power at best... without the gold and silver from the west and the oil in TX, I don't see how the US would ever become either an economic or military power... nobody would want to ally with us in any world war...
Well we are not talking the complete Oregon terr. here. the most likely option would be that the US simply is forced to accept the British positon regarding the border at the Columbia River...That leaves eastern Washington and all of Oregon state with the US and simply adds the entirety of Puget Sound to BC ( an area that is sparsely settled at best at this point) the entirety of what would be Washington is only a bit over 1000 I think.
This would hardly be punative to the US. Even if Britain decided to redraw the Maine border in their favour that too is hardly punative to the US and has only recently been settled in US favour. Think of this as a fight between relatives as you will...one side comes out ahead and the relationship will be decidedly frosty for a number of years...but the fundamentals for the Transatlantic relationship between Britain and the US are still there. The US may lose but even with the above border modifications of what were disputed terr. to begin with they will get off quite lightly and there is no reason to think that the two will after an appropriate period of cooled relations will move back on a path of convergence of interests once again.
If Britain were to feel a need for punative measures...then it will depend on the course of the War of course and its length. the most probably punative measure would be some form of reparations...or to lower the western border to the 46th parallel. Essentially for a parrallel border that extends from the mouth of the Columbia across the continent to the UP of Michigan.
Other than a few US Army forts and a whole lot of Natives there are also few if any settlers in the region. This would be in lieu of reparations most likely.
Even if the Brits felt a need for some symbolic punative measure this would not be a hardship for the US at that time as the potential of the Mesabi has not even been discovered yet. All it really does is put the entirelty of the Red River valley within the British sphere ( a historical claim but surrendered in 1818) so really they are only getting back lost terr that they initially claimed.
This is highly unlikely and really depends on the course of the War and wether the Brits would feel that a punative measure was indeed required...
It would depend on who started the shooting and under what circumstances, and how well the Brits are able to rally the Natives to their position.
Even at this, the relationship might remain frosty for a somewhat longer period..but by the 1890's I suspect that they would once again be seeing themselves on a path to convergence of interests once again.