Despite the War of 1812 being a stalemate it didn't solve any of the grievances between the two countries. The Americans still felt that the British were acting as agent provocateurs with the native population. The biggest problem the British had was the American border claims of British North America (modern day Canada).
The British still supported the native population but after the 1812 war it stopped supporting them activley as the British politicians saw the American market as having massive potential. That not to say they didn't trust each other as the they still didn't. There were a number of powder keg moments which nearly led to renewal of hostlities such as the Oregan border dispute and the Trent Affair during the American Civil War.
What started the budding friendship was when Britain and Venezuela disputed the boundary between the latter country and British Guiana in 1895, President Grover Cleveland pressured the British into agreeing to an international arbitration. They issued its verdict in 1899, awarding the bulk of the disputed territory to British Guiana. Despite this setback for the United States, it showed that standing with a Latin American nation against the encroachment of the British Empire improved relations with the United States' southern neighbors. However, the cordial manner in which the negotiations were conducted by the United States also improved diplomatic relations with the United Kingdom.
This led to the Great Rapprochment which is a term that was used to specifically describe the convergence of social and political objectives between the United Kingdom and the United States from 1895 until World War I. The differences that had separated an industrialized United Kingdom and an agrarian, anti-imperialist United States where Anglophobia ran high, rapidly diminished in the decades preceding World War I.
The most notable sign of improving relations during the Great Rapprochement was the United Kingdom's actions during the Spanish–American War. With the onslaught of war beginning in 1898, the British had an initial policy of supporting the Spanish Empire and its colonial rule over Cuba since the perceived threat of American occupation and a territorial acquisition of Cuba by the United States might harm British trade and commerce interests within its own imperial possessions in the West Indies. However, after the United States made genuine assurances that it would grant Cuba's independence, which eventually occurred in 1902 under the terms dictated in the Platt Amendment, the British abandoned this policy and ultimately sided with the United States unlike most other European powers who supported Spain.
However, even when WW1 broke out it was not clear who, if any the American would join. There was a powerful German lobby and there was every chance that the Americans would side with the Germans. Howevwer, as evidence of German complicity in public in conspiracies in and against the United States such as the Zimmerman Telegram, it became more obvious that American public opinion was becoming more influenced to the prospect of joining World War I. When the German Empire responded in 1916 with a submarine blockade of the United Kingdom and the sinking of the RMS Lusitania by a German U-boat, it led to a protest by the United States and a strong sense of anti-German feelings among the American people.
With the entry of the Americans into WW1 it brought the relationship from a rapprochment to what is called today 'The Special Relationship'