I'm not in London but up north, and it's the same here. In my experience you have to wait until the next connecting train, or buy a new ticket and catch a suburban train from a different company that takes 3x as long. This one is sometimes late and will skip your station. I've missed work (seminar teaching, short and time-sensitive, so not just a case of arriving late and staying late) because of cancellations and trains going through my station rather so they can be on time to Leeds or wherever.