Expanding Gatwick won't provide the same benefits as expanding Heathrow. The demand is at Heathrow, so expanding Gatwick would require waiting for demand, airlines and their infrastructure to shift from Heathrow to Gatwick. Plus, the logistic handling capability is already at Heathrow, because more than 200 of the top 300 companies in the UK already have headquarters within 25 miles of Heathrow.
Expanding regional airports also will have little effect. Passenger demand is greatest in London, so it would make no commercial sense for an airline to move the bulk of their operations to a larger regional airport - even if there was space for more flights.