Well, a few weeks back, PBS aired a documentary on the War of 1812. One point they made is that the British had also essentially forgotten that war until the 1959 song "The Battle of New Orleans" got a good deal of airplay. The British, of course, were more concerned with Napoleon at the time. On this side of the pond, during my time in school, the war became famous for the Francis Scott Key poem that became our national anthem and not much else. It was never seen as a resounding victory for the US: much of what we went to war over was resolved before the fighting began, and its most famous battle, teh aforementioned Battel of New Orleans was fought after a truce had been declared. This all due to the slow means of communication/travel at the time.
I also agree that if the British had not been focussed on the rest of Europe, the fledgling USA would have been reassimilated into the empire and any remaining "patriots" and founding fathers tried for treason. Now there's a plot for Harry Turtledove to work on! (Given the media's and general public's fawning attention to the royals over the last several years, I sometimes wonder if we haven't rejoined the Commonwealth.)