The UK could renegotiate air services agreements and dictate that Airlines fly to Gatwick or Stansted rather than Heathrow.
All been tried before in the 1980's - it didn't work then and it wouldn't work now. BA moved all its Canada, Spain and various other routes from Heathrow to Gatwick. The reciprocal airlines simply stayed put at LHR and made a killing until BA had no choice but to move back again. Any move to oust foreign carriers from LHR could be expected to be met with reciprocal action or trade sanctions against UK carriers at the other end of the route.
Governments getting involved (interfering) in what should be airline business decisions rarely turns out well.
The current supply versus demand for LHR slots already ensures that new airlines or extra frequencies have to go to other airports - unless the airline concerned has a bottomless pit of money to buy the slots it wants at LHR from another airline (usually its an airline going through hard times and selling off the family silver !). Hardly fair on the smaller guys - but that's the way it is in the absence of LHR R3.