london city is close to heathrow and that works
Nowhere near as close as Northolt is. City is in East London, Heathrow is beyond West London in Middlesex.
So you expand Northolt.
Your customer airlines have a split between those coming to London and those connecting to partner airlines. STAR ALLIANCE under one roof in the new T2, BA in T5, OneWorld in T3.
However now we have to choose who gets sent to RAF Northolt which is a different airfield. Who gets kicked out? Do we send them domestic? If so, connection times are screwed and they'll use the competition in Europe.
Give one of the other airports the new runway, if its needed at all, and improve the infrastructure and transport links to them and watch heathrow's problems disappear..
That's incredibly simplisitic. BA tried to expand into LGW, except every time they moved a route to LGW, the passengers stayed at LHR and flew with someone else. The London airports have distinct roles of their own. Giving LGW another runway is nothing to do with capacity at LHR.