To answer the OP's question. You run to your airline and hope to get on another flight later the same day. If you are lucky you might get put on to another airline's flight (don't count on that). If your luck is poor you might have to spend the night at LHR and start over the next day.
To make things easier for you, go very easy on carry on stuff. A small carry on bag is to your advantage. Be polite and hope for the best.
I have to tell you that I am now avoiding most British airports for very good reason.