Usually you change in London, but sometimes you might instead change at a US gateway (such as Boston, Chicago or Atlanta)
You fly direct only on charter flights... such as those for Sea Princess sailing from Barbados.
If you change planes in the uS, you will need to go through US immigration, collect your bags, go through customs, recheck your bags for the next flight and then go through security. However, when you arrive in Miami you don't need to go through immigration a second time.
If you change planes in the UK, you may need to reclaim luggage and recheck it, depending on the airlines. When you check your luggage in, ask the check in agent where your bags have been "tagged through" to. Often your bags are tagged all the way to Miami. On disembarkation in Miami, expect a queue for immigration... at least 1 hour... before you claim your bags.
Personally, I prefer Delta airlines via Atlanta... the seats seem bigger and the cabin crew are just great... plus the beer is chilled.