it's not going to be as easy as hitting the TOGA button and taking the shortest route to that nice nearby airport and landing there. All of the normal flows of traffic will be disrupted.
Ultimately it comes down to "Command" and awareness.
Any airfield, be it 1, 2 or 3+ RWs could "close" with no notice. Fuel policies
allow (but do not require) you to re-arrange your flight to land with "(just) above Reserve" without telling anybody (e.g. ATC) nor having a "Plan B" (Diversion).
I have my "ways" of dealing with such situations, I am sure others have theirs. They vary from informing ATC as soon as possible if a GA will result in either a PAN or Mayday. I suspect (but do not know) that if an aircraft was inside 5NM for LHR, and had clealy notified LHR that "in the event of a GA I will be declaring a Mayday for Fuel", then on Friday they would have been allowed to land.
In the event a short of fuel late GA happens, I will have briefed with my colleague the "bottom" line. This might be 7700 and direct track to 8NM final at XXX, whatever ATC say.
Others may, or may not, disagree. But the other side of the coin - working without thought to minimum fuel policies, and then expecting the normal ATC system to cope with mass diverts and have everybody on the ground with > Reserve, I think is "optimistic"