And not everyone in the UK had interest
"I think you have no right, from the state of war, to demand any concession of territory from America... You have not been able to carry it into the enemy's territory, notwithstanding your military success, and now undoubted military superiority, and have not even cleared your own territory on the point of attack. You cannot on any principle of equality in negotiation claim a cession of territory except in exchange for other advantages which you have in your power... Then if this reasoning be true, why stipulate for the uti possidetis? You can get no territory: indeed, the state of your military operations, however creditable, does not entitle you to demand any"
Duke of Wellington, November 1814
Now add on the US holding a good chunk of British North America
Enough of those in Power recalled how expensive the ARW was to fight, and there was already a problem with desertion in the British Army, over 70,000 cases during the Napoleonic Wars, and now new orders to fight in America after Spain and France would not be looked well on.
Wellington's whole letter though, is hardly saying the British have no right to demand territory. He's merely pointing out that uti possidetis is a pointless claim when the British are not properly standing on the territory they seek to claim directly. Indeed, most of his foibles come from the British having direct control of the Great Lakes. In the same letter he states "If you had territory, as I hope you will soon have New Orleans, I should prefer to insist upon the cessation of that province as a separate article than upon the uti possidetis as a principle of negotiation"
There's been a lot of misunderstanding of exactly what Wellington was saying in that letter I think. Had the British been entirely successful in their 1814 campaigns, rather than partially, and been standing in more than the putative state of Maine, Wellington is clearly saying that demanding the cessation of territory is a good idea. However, insisting on the mentioned uti possidetus or "as you hold it" as Liverpool was insisting, seemed unwise to him.