There is some discussion between the diocesan authorities and the district council with regard to who is responsible for making sure the monuments and memorial stones in the churchyard are safe. In due course, either the district council or the P.C.C. will have to obtain a faculty and lay flat any memorial stones which are unsafe. This will have to be done for reasons of health and safety.
In the meantime, the next of kin have the opportunity to make sure the monuments are made safe. If this is done by a professional stonemason, there will then be no reason for the monument to be laid flat.
There is absolutely no way that either the local authority or the church authorities can take responsibility for repairing monuments. That is the sole responsibility of the next of kin.
Ann Andrews has provided extensive help with searching memorials in the churchyard on her website. There is a map and an index of memorial inscriptions. You will find more information about genealogy and war graves on the ‘links‘ page.