There is no such thing as a woodland protection order. I suspect you might mean woodland Tree Preservation Order (TPO.)
A landowner wishing to fell trees subject to a woodland TPO would have 2 considerations. (1) Is a Felling Licence required from the Forestry Commission. If the volume of timber were to exceed 5 cubic metres per calendar ¼, and assuming other limited exemptions did not apply, the answer would be yes - they do need a FL. (2) if they were planning to remain < 5 cubic meters per ¼ then FL not required but a planning application to LPA would be. % m2 isn't much in reality..
Part of the FL application process would be to declare the presence of the woodland TPO if known. Once declared, the Forestry Commission would liaise with the LPA to ascertain any conditions which might be applicable in relation to replanting.
The PC attempting too influence the public would have no impact whatsoever on the process of granting the FL - in fact, there isn't even a mechanism for the public to submit comments to the FC. There could be if it is a planning app but completely impracticable anyway - the LPA will disregard whimsical public comments.
Ownership and ability to grant or withhold consent for works on a village green is a separate issue and there is insufficient info to comment purposefully on that.