Who says they need to increase their cost base? In all honesty, how much money would it cost me as a passenger if they did? Diversions are not handled for free, they pay landing fees, ATC fees and parking fees and all the rest of it. The major cost is to the diverting airline. A well run airport can make money from this misfortune, a pared to the bone operation misses the opportunity.
There is no need for any smaller airport to make any infrastructure changes to handle LHR diversions. MAN, PIK and a few others are Code F and can handle the largest aircraft, the rest can be accepted where there's space.