How long did it the United States to forgive the British for their actions in the American Revolution and War of 1812?

Not very long at all.

Britain was the pre-eminent global economy at the time and the United States, as a relatively small nation with little to no industrial base, needed a trade relationship. After the end of the Revolutionary War a trade deal with France failed to materialise and so American merchants returned to trading with Britain (much to the annoyance of the French). This trade grew pretty big until, as part of the build-up to the War of 1812, the United States enacted a disastrous embargo.

Again, the fall out from war didn't last very long, and US merchants quickly returned to trading with Britain once the war was over. Money, after all, was more important than pride. Not trading with Britain then would be like refusing to trade with China and the USA combined today.

Away from trade, the USA was isolationist for most of the 19th century, not wanting to get drawn into imperial wars, and Europe didn't care that much either. They'd all turned their attention to Asia and Australasia by this point. So in this regard, forgivness as such didn't really need to happen.

A really good book in this is Gordon S Woods "Empire of Liberty", which is part of the Oxford History of the United States series. It's a bit hefty, but you'd only be looking at a couple of chapters.