LHR and LGW crews operate entirely separately, on separate contracts and agreements.
LGW crew check in, brief and operate from LGW and the same applies to LHR.
In the event that a crew from one base covers a flight from another (due to disruption, crew shortages etc) then they have to sign in and brief at their normal ‘home’ base and they are transported by the company to the airport they’re operating from.