I believe the best solution would be to close LHR after building a new airport from scratch. An additional runway would just make the place even more difficult to operate from. BA has to compete with the very best in the world, but it has to do so with the the anvil of Heathrow tied to its back. With BA and the UK in mind, I believe the solution should be radical. The first thing to do is to accept that LHR will have to close in its current form (or maybe altogether - letting it stay open is like keeping 97 year alive on a life support machine) and that it's replacement will be elsewhere. Why? Because if you travel from say BPK where my parents live, you have to give yourself a buffer of something like two hours to allow for congestion. Parking will add another 90 minutes or so have you seen his much that costs! To travel from the City or the West-end, which many do takes well over an hour. But the rail connection from Paddington takes the biscuit. Prices start at £21 for a single - but how do you get to Paddington? Basically, it's location is appalling and it has insufficient room to perform.
I believe the best place would be somewhere in the north west, say in the sea near Liverpool or Blackpool. Then, this airport should be planned as a multi-site complex with as many processes as possible handled offsite.
Radical solutions to check-in, parking, transfer, security could be applied. Like there would be no on-site parking. Ditch the shopping complex and have the transfer lounges remotely located as well. Have maintenance performed offsite. Place the main (remote) terminals over a rail line (that is not a spur or branch line). Have on-train (and on-bus) check-in and "security" performed off-site as well. Then the further growth of the airport will not restricted by the land.
The runways should be placed sufficiently far apart to enable simultaneous CAT III landings. Other innovations I would suggest would be no cul-de-sacs, air-co/cabin heaters, fuel and FEP on every stand with one man TBL tugs as a default. And while you are at it, make every ground vehicle with the exception if the RFF vehicles, electric.
When open, chop up Heathrow and turn it into a small regional airport for commuter planes for people who wish to fly to Hounslow or Hillingdon (or better still, close it altogether).
But I'm a lucky person, because this will not happen. The punch-drunk, terminally wounded, ageing pub fighter called Heathrow gives me a job. Because it is so poorly situated, so unpleasant to use and has so little capacity, many people choose not to use it. Instead, they fly from other regional airports to other hubs, one of which is home to my employer.