Errm...
The American system isn't really that different to the British one... whichever party has received the most first-past-the-post votes gets their candidate (95% of the time; hung parliaments and emergencies are the exceptions).
As far as I'm aware, if an American President resigned, died or was removed mid-term they'd end up with their Vice-President, much like we ended up with our 2nd in Command, rather than calling a snap-election.