A friend of mine missed his outward flight which included a connection in Heathrow. Even though if he took the next flight to heathrow, he would still have made his connection on time, his whole itinerary was cancelled. Luckily the airline charged him a 'no show' fee of €50 and allowed him to rebook which didn't cost anything on top of the original price.
But be aware that this was only because his ticket allowed this. If you have a discounted ticket, a lot of airlines do not allow you to change/cancel and tough luck if you miss your flight.