I’d refer you to:
Paul Clayden (2008)
The Law of Parks and Open Spaces
Sweet & Maxwell ISBN 978 0 7219 0552 5
Which states that s10 of the Open Spaces Act 1906 affirms that a local authority may exercise control over a public space and that s20 defines a public open space and that s11 requires that a faculty must be obtained prior to undertaking any work
Interestingly, s11 makes special provision for the removal of monuments. Headstones, tombstones or monuments in a disused (closed) cemetery may be moved but the authority (including a PC) must at least 3 months beforehand prepare and publish a statement of sufficient detail and description of particulars and make the statement open for public inspection. The authority must also advertise, on 3 occasions, in a newspaper circulating in the area the intention, the existence of a public declaration of details, where and when it might be inspected AND also post a notice on the door of the church and send copy by mail to any known near relatives.
If the ground is consecrated, the authority must wait at least 1 month after the date of the last advert and then apply for a faculty from the Bishop - no monuments may be moved until such licence or faculty has been obtained
As can be seen, there is considerable moral, legal and practical responsibility associated with such proposals and they should not be entered into lightly, frivolously or without proper care.
This desire to move or lay down headstones has a tendency to arise when a well intentioned but often poorly informed pseudo Elf & Safety character propagates fear mongering and risk over inflation.
Proceed with caution - the referenced book is available from Amazon and SLCC bookshop at less than £50 and would be a very sound investment for any PC