If the reason for Heathrow expansion is to increase connectivity, then in theory it shouldn't matter whether the expansion is at Heathrow or, say, Aberdeen.
If the reason for Gatwick expansion is to increase connectivity, then it will need a whole bunch of airlines to redirect inbound and outbound services from Heathrow to Gatwick - which certainly hasn't happened in the past. Very few airlines, since PanAm, have shut down at Heathrow and moved to Gatwick and I can't think of any that prospered from a Heathrow > Gatwick move.
On the other hand, if the reason for expansion at either or both is to facilitate an increase in O&D traffic into the UK, then the obvious answer has to be something, somewhere, more central inside the UK and which would probably be 10x cheaper and easier to realise. Not too sure where and how, though, so expansion at Stansted to the north, perhaps Oxford to the west, and Southampton to the south would appear to offer the travelling public huge advantages at significantly less cost to the UK Government than pouring £billions of public money into either of Gatwick or Heathrow.
Well, that's how I see it.