The ownership of war memorials is a complex question.
This is a complex question, explained in detail in a document from War Memorials Trust.
In summary, the majority of war memorials were paid for by public subscription through a committee. The ownership of a memorial can only be established if a formal transfer of ownership occurred after the war memorial was erected. In many cases this is not possible as the committee that created the memorial was dissolved after its work was done.
For war memorials in churches and churchyards the church has traditionally maintained them (as they do other monuments in the building). Before work is done to a memorial in a Church of England Church a faculty will be required.
For memorials not in churches when the ownership can not be established the War Memorials (Local Authorities' Powers) Act 1923 enables local authorities to carry out work to war memorials whether or not they own them.