Nevertheless you clearly have a serious problem. Your first contact must be parent-governors, whose numbers have recently been increased in almost all schools. As a church school you will have elected representation and also (by law) parents among the church-appointed group.
It is hard for these few to stand out against a large group of church appointed governors if the latter are united, but you must impress on them that their special duty to parents, especially to communcate properly on any non-confidential matters, and that they must be brave in speaking up and voting for the chair they want. Governors can remove a chair mid-term if enough of them are dissatisfied, subject to strict procedures to ensure fairness.
Your next hope is the annual meeting with parents. This is required by law, and although attendances are often poor, parents can use these to discuss any matter of concern and to hold governors to account.
If this fails and your education authority really won't respond, the only other hope is to write to the Secretary of State, under section 496 of the 1996 Education Act, which allows appeals against unreasonable behaviour by a governing body or education authority. However, very few appeals have been allowed because the courts say that only outrageous behaviour justifies one statutory authority over-ruling another.If your dissatisfaction with your governors centres mainly on personnel issues, there is often no alternative to secrecy. There may be a need to protect the privacy of an individual or to ensure enough governors are kept unaware of issues so they can hear appeals.