I am not so sure that the commercial case is made for LHR, Take New York it has 3/4 a/p's equally spaced around it thus allowing road traffic to get into Manhattan from different sectors, If you expand LHR you exacerbate the road access problem not ease it. Plus the M25 is a key road link between the Channel Ports and the North, If this traffic goes anti clockwise it gets snarled up on the Thames Crossing, always very busy or it goes clockwise past LHR again currently gridlocked in the peak times. Dig it all up and the country could suffer horrendous loses. Air traffic is important but road freight is more important to the UK economy.
So yes an improvement at LGW would make more strategic sense as it would allow a different point of entry into London and would be a much shorter route for air freight seeking to use the Channel tunnel, both in or out