If an individual train is cancelled you can use your tickets on a later train. (In most cases if you get to the station early, and see that your scheduled service is cancelled, you can travel on earlier train too).
If a problem with one train service on your journey means that you miss your scheduled connection, you can travel on a later service.
If a line is completely closed, with no alternative transport laid on, you can travel on the first day when the service is restored or get a refund on your tickets.
Further, if you're delayed by 30 to 59 minutes you can claim back 50% of the price of a single ticket or 25% of the price of a return ticket. If you're delayed by between 60 and 119 minutes you can claim back the full price of a single ticket or 50% of the cost of a return ticket. If you're delayed by 120 minutes or more you can claim back the full cost of both single and return tickets. (Those detail apply across most rail operators but individual franchises might have variations).
Full details can be found in the passenger charter for the relevant rail company. Look for the link to it here:
http://www.nationalrail.co.uk/times_fares/ticket_types.aspx#Refund