Did the relationship begin to mend as soon as the peace was signed, or did it take time for the nations to change their attitudes toward each other?
I'm not an expert in the period, but from my somewhat limited reading of the period, I would put the beginning of the shift from foe to friend in the period right after the War of 1812. Remember, the US and UK were fairly adversarial all the way up through 1815, they didn't just become chums after the 1783 Treaty of Paris.
In 1818, the US and UK signed the Treaty of 1818, which was a pretty huge treaty with land concessions from both sides in order to establish the border on the 49th Parallel, in addition to a few other things.
A few years later, you had the announcement of the Monroe Doctrine, with the support of Great Britain. This is a very significant event in that it marks the first time that the UK and US find their interests truly aligned. It served the foreign policy of the United States to declare the Americas essentially off limits for European intervention, as it helped establish a degree of international clout for the US. It also served the economic needs of the UK, as foreign markets in Central and South America would be locked up if they were to return to colonial status. The UK had heavily shifted its economy towards the manufacture of finished goods over the previous centuries, especially with the advent of the Industrial Revolution, so what the UK needed first and foremost was markets to sell its goods.
At this point, the US and UK found it better to work together than as adversaries. The US didn't have the power projection capabilities to truly enforce the ideas of the Monroe Doctrine on their own, and thus relied on the promise of British naval power to intercede on behalf of the US if any European country were to violate the Monroe Doctrine.
This decade following the War of 1812 is what I would really identify as the shift in US-UK relations. The two sides began to settle their disputes more amicably, and found themselves in a mutually beneficial policy situation that would serve to get the two countries closer.
The only specific book that I've read that I can think of off the top of my head on this issue is Promised Land, Crusader State: The American Encounter with the World since 1776, by Walter A. McDougall. There are probably contributors on here who can attack this from a more Anglo-centric viewpoint and add more to the discussion.
FYI, you'll find some previous posts on this topic in the FAQ