It's 1814 and the War of 1812 just ended without the reclamation of the North American colonies by Britain.
Fast forward to World War I a century later. US and Britain are allies against Germany.
Was there a specific point at which the British said "OK fine, you win, let's be friends" or did relations thaw over the course of a century, leading to the UK and US' statuses as each other's closest ally?
If we are talking about when the US and UK began normal, neutral relations, it would be at the end of the War of 1812. The American Revolution officially gained independence for the colonies, but after The Treaty of Paris (1783) was negotiated and signed, issues still remained including British forts being maintained in the US North West Territories, support of Native tribes, and the high-seas seizures of American sailors and their forced conscription into the British Navy. These issues drove the US to declare war on Great Britain in 1812, and two years later all of these issues were resolved in 1814 with the Treaty of Ghent, as well as settling border disputes with the British territory in Canada.
World War 1 was actually the point where US and Britain laid the foundations for the close relationship that both nations have today. Between the War of 1812 and World War 1, the US and Britain maintained a neutral relationship. During the Civil War, Britain conducted trade with both the Union and the Confederacy, and also considered an invasion of the North in just the early stages of the conflict. But this was the closest US/UK relations came to deteriorating again.
When World War 1 began in 1914, the United States had decided to remain neutral while Britain aligned itself with France and Russia against Germany and Austria-Hungary. From 1914 - 1916, the US traded goods with both Germany and the UK, although mostly with the UK, including selling arms and ammunition. When Germany opened up its unrestricted submarine warfare policy, it pushed the US to enter the war on the side of the British, notably as a result of the sinking of the passenger ship the Lusitania, which carried passengers and arms and was sunk 100 years ago yesterday.
Both nations were victorious in 1919, though they were all but allies until the US entered World War 2 in 1941. Trade was flowing between the nations in the interwar period, and they both negotiated an equally higher ratio of naval ships compared to Japan at the Washington Naval Conferences. US and UK relations were strong at this point, but it took World War 2 to seal the close relationship that exists between both nations today.
Hiya, you may be interested in a few earlier posts on this. Not discouraging more discussion here, but this can get you started: