LHR is the right place for the extra runway. The other London airports have available capacity across the day (even if they are close to full at peak times). I'm a big believer that if the demand was truly there at LGW/STN etc. (as it is at LHR), then airlines would be taking slots even at 'unfavourable' times. I'm also a big believer that a successful airport hub relies on interconnectivity—concentrating flights in one location maximises transfer opportunities and makes more destinations viable, increasing O&D traffic too. Splitting flights across multiple airports fragments connections and routes. It clearly works to a point, but 'divide and conquer' isn't sensible, IMHO.

I may be misremembering, but I believe an IATA/ICAO rule (somewhere...) dictates that at least 50% of any new runway capacity added at airports must be offered to new airlines. That would open the door to easyJet and would (surely) risk a big dent in BA's short-haul loads.

I'd heard on the grapevine that to make a base at LHR economical, EZY would have needed a good number of slots—more than were available—so it never happened (undoubtedly with some lobbying from BA/their QR overlords). From what I understand, EZY have taken a look at Terminal 4 on a few occasions in the past, but clearly to no avail.

Also, I don't believe BA have owned Waterside in quite some time, they sold it to a landlord and rent it back. I'm confident that their fiercest objection will be opening up their LHR monopoly to more short-haul competition.

*EDIT* - found the rule I was referring to: Worldwide Airport Slot Guidelines (WASG), jointly published by IATA, Airports Council International (ACI), and the Worldwide Airport Coordinators Group (WWACG). According to these guidelines, 50% of any newly available slot capacity at coordinated airports is allocated to new entrants.