In 1815
1. The French had just revolted again, and the Saxons hadn't been any too dependable at Waterloo. The British weren't sure what was going to happen on the continent. They decided to let the situation settle down a bit before causing any more excitement. Sweden might have joined in, Italy might have joined in, etc. No telling what would happen.
2. The Austrians were settling the fate of Europe. The British wanted to be there to make sure they got their share. If not, Britain could be cut out. Concentrating on diplomacy was making sure you got rewarded for the last twenty years of sacrifice. The other European powers weren't interested in Napoleon.
3. The national debt was sky high as a proportion of the GNP. The British had just come off twenty years of war. They were tired and not any too certain of the loyalty of their own troops. Even if the troops didn't revolt, they might just refuse to fight. Britain didn't have a large army to defend it's borders, so it didn't have a large army to defend it's government.
4. The manufacturers didn't want any more high food prices. If the corn laws and commerce raiders kept the price of food high, then they had to pay the workers more to induce them to stay in the cities instead of going back to the pastures and plowing them up for potatos and beans. A longer commerce disrupting and raiding war with America was too much for them to bear.
5. The last time they fought a war with America, forty years ago, substantial portions of their army changed sides. Since then America had gotten considerably larger and more powerfull. More troops might decide that since America was going to win anyway it would be safe to desert and go live in America. Their army could simply fall apart and be defeated. My ancestors were among the British troops that deserted. Well, Hessians, anyway.
6. Cotton was getting to be important as the crop grew geometrically. Disruption of cotton imports was also a problem. This directly impacted the manufacturers. Wool was made in Britain, but cotton was a new crop and cotton imports essentially ceased after the war and embargo on trade. The price of wool going up was matched by the price of woolen cloth going up.The problem of cotton was that there wasn't any to sell at any price.
7. The sugar crops of the Caribbean were not as important anymore because of the improvements in the yield and quality of sugar beets. Louisiana was not important as potential sugar territory. Nor was it important for food production. Some corn came down the Mississippi from the Ohio, and that was also embargoed, forcing the planters to raise their own corn in the Caribbean, further increasing prices.
8. India was an important and profitable market. If the British concentrated on India, the French, Portuguese, and Dutch allies against Napoleon might sieze the opportunity to regain lost trade from the British from the larger part of India that wasn't under British dominion. Trying to grab America would involve losing India.
9. Australia was an important source of future trade. As far as potential for wheat and wool production was concerned, why bother with America. There were lots of other markets and lands that weren't so ferociously defended. South Africa was wide open. So was New Zealand. Screw America, we have other concerns was the British attitude.
10. Remember, the whole impressing the crews thing was to fight Napoleon. Without a war, why bother enslaving American sailors? The British didn't need them anymore. To build a navy to fight America to defend the British right to conscript American sailors to build a navy...
So to answer the question "what would have happened if the United States had lost the war of 1812", just assume that the British don't avoid being sucked into a continued war and see everything go wrong.
The defeated British watch as the Dutch, Spanish, French, Russian, American, Italian, German, Swedish, Irish, Danish, and Portuguese leaders meet in London to argue the fate of the Empire. Indian and Turkish and Chinese leaders are pointedly not invited.
Oh yeah, one other thing. 1816 was known as "eighteen hundred and froze to death". Tamburu had sort of cleared it's throat off in Indonesia and we had a mild version of volcano winter. If the British government was involved in a war that prevented food imports they might really have been overthrown by their people whether the rest of Europe went after them or not.