This relationship didn't build overnight. Anti-British sentiment continued on throughout the Napoleonic Wars, because of propaganda that was spread in the United States media, including incidents of impressment of American sailors on merchant vessels, sale of weaponry to Native Americans in the west, a push to protect citizens living on the frontier in Tennessee and Kentucky, etc. The War of 1812 was fought against the British for many of these reasons, and land greed many Southerners exhibited (War Hawks like Henry Clay, Felix Grundy and John C. Calhoun wanted to take much of eastern Canada for the United States).

What stopped the US from entering another war with Britain for as long as they did was the tax system that was in place in the early US. There was no income tax or sales tax at the time. The only federal taxes in place were excise taxes (which were unpopular) and tariffs (taxes on goods imported into the country.) Roughly 90% of imported manufactured goods came from Britain, where the Industrial Revolution had already begun. Because tariffs made up most government revenue, that meant that almost 90% of the government's money came from that trade. If we went to war with Britain, we could be decimating our annual revenue.

After the war of 1812, we have had a more positive relationship with Britain mainly due to increasing the size of the US military, a booming economy, with a dash of good politics on behalf of Secretary of State John Quincey Adams. With all of the Wars that Britain had fought - Seven Years War, War for American Independence, Napoleonic Wars, War of 1812 - they were deeply in debt and not sure how to solve that problem. During the American Revolution, raising taxes on their citizens was a huge problem and the British government was becoming wary of collecting too much from their people all at once. They were rather looking for solutions to build their economy steadily over time so that taxes could be raised without putting too much pressure on the populous. This sentiment led to peaceable agreements with the US, like the Rush-Bagot Agreement to demilitarize the great lakes and capitalize on their economic value for both countries, the convention of 1818 to set the borders between the US and Canada, and the Oregon Treaty, which would peaceably divide the Oregon Country, which had been shared for a long time until both parties could figure out what to do with it. The end result there was to divide it in half (North/South) and share fishing rights in major rivers. The relationship was still tentative, but more peaceful that it had ever been.

It wasn't until totalitarianism threatened the nature of democracy in the 20th century that the US and Britain really got together behind shared ideals of democracy and western ideology that made them besties.