Faculties and Planning Permission
Although Ecclesiastical Exemption means that Churches do not need to get Listed Building Planning Permission for internal works, if you propose work which will alter the outside of the building (eg an extension for a Hall or a Toilet or even floodlighting), you will need Planning Permission.
For some works you will also need to involve Building Regs (eg a new building, work to drainage, a new boiler).
If you want to fell, lop or top a tree in the churchyard or on other church land there may, as well as the need for a Faculty, be a need to contact the relevant Local Planning Authority if the tree is the subject of a Tree Preservation Order or is within a Conservation Area. As usual, the advice is to contact your Archdeacon.