I like that idea, befree. Instead of assigning a direction for each airport have each airport assigned a continent. For example,
Heathrow -- to/from North and South America
Gatwick -- to/from Asia and Australia
Stansted -- to/from Europe and Africa
Luton -- to/from GB & I
Both Stansted and Luton would need major upgrades, but it shouldn't be too much of a problem because there are not as many NIMBY's up there.
Make transferring from one airport to another as seemless as possible. Have the high-speed railway cars secured so that it wouldn't be necessary to go through security when reaching the airport.