I think the BBC would probably run cap in hand to the MOD to pay for a new transmitter in that case. The BBC transmitter is getting on for 70 years old, once it fails repairing it will require custom made equipment or complete replacement - both of which will be very expensive.
One other curious issue with radio 4 LW is that it carries a data sub-carrier for the National Radio Teleswitch. Basically the signal that controls the Economy 7 and Economy 10 heating systems. If Radio 4 LW goes off-air any renaming E7 and E10 systems will just stop working.
One crazy proposal that went nowhere was for the BBC to move Radio 4 LW to RTÉs Clrakstown transmitter in Ireland. That used to be the Atlantic 252 transmitter (remember them?) but has been carrying RTÉ Radio 1 since that venture failed and RTÉ really wants to shut it down. The transmitter at that site was built in 1998 which makes it a very young in radio transmitter terms.