LessThanSte - Some good points
Regarding weather, it's interesting how it can vary so much at the London airports on a winters day, particularly fog and snow. From my experience Heathrow being west of London tends to get less snow than Gatwick and Stansted with Stansted more likely getting snowfall being over 300ft elevation and it's further east location influenced by north/north east cold winds bringing showers inland from the North Sea.
Gatwick is situated in a valley with a river running nearby so perfect for lower temperature overnight and formation of fog yet at the same time Heathrow and Stansted are clear with higher temperatures. Although Stansted does suffer from fog occasionally in winter in certain weather patterns due to it's rural location and elevation and often Gatwick and Heathrow are clear at the same time.
Another issue with single runway is an alternate in case of diversion. For example if one of Heathrow's runways experienced an unplanned closure the alternate is the other runway but with a single runway airport the alternate would be to another airport, provided of course the ground handling and parking capacity at the alternate is available. I believe with single runway airports aircraft may uplift more fuel at the origin so in the event of a diversion they have sufficient fuel for the alternate airport.
Another issue with the very busy London airports, which includes Luton, if there is an unplanned closure of any of these airports runways then a situation quickly develops where many aircraft are looking to divert at very short notice to other airports and in the space of a few minutes this can cause significant increase in ATC workload. Also some of these airports may not be able to accept these diversions for reasons already mentioned.
Altogether a very interesting and complex subject