on the next available train. the operator should cover you even if you are using split tickets, provided that you allowed the minimum connection time at the split point and that you travel on a train run by the same operator, not just the next train. note that if the delayed first leg of your journey was by tube, you are covered if the tube and rail journey are on one ticket, but not if they are separate tickets.
if you travel on the wrong train with an advance ticket for any other reason you are liable to pay the fare that would have had to pay if you had bought 