On BA, I've fortunately never been in this situation. But what happens if you miss a flight, maybe due to traffic or whatever other reason, but for some reason that has nothing to do with BA. What exactly do you do? I'm actually clueless as to what your actual options are at that point. Have you lost the entire ticket and have to rebook a new one at same day prices? Can the check in staff help you?

Secondary question, what if you miss your flight once you've already checked in and past immigration? Who do you contact in that sort of situation? Assuming not in LHR so not any BA staff around