In 'public' cemeteries, churchyards and graveyards, a headstone is generally paid for by the deceased's family, occasionally by public subscription. When repairs are needed, the family has to be traced, but if that's unsuccessful, the headstone can be removed. No-one unconnected with the family can do anything to the headstone/memorial, such as have it cleaned or re-erected; to do so, you'd have to have permission from the family.
"Whilst the cemetery or graveyard owner has a health and safety duty for the site, the individual monuments are privately owned and the descendants of the original grave owner (if they can be traced) are technically responsible for maintaining their memorial in a safe condition. Often monument owners cannot be traced, but where repairs are needed in the interests of preserving the monument and the amenity of a cemetery, the cemetery owner and conservation groups may step in to help with the costs."
For example, here we have a memorial to an ABW veteran. Presumably the family can't be traced, so if it should be at some point be judged a danger, it could be removed. As I took the photo getting on for 10 years ago, it might already have been removed.
My question is who pays for headstones and memorials in military cemeteries? The families of those interred there? Okay, that's two questions.