The Church of England website, which includes details on planning a funeral and the funeral service, notes:

"A funeral is used to mark the end of a person's life here on earth. Family and friends come together to express grief, give thanks for the life lived and commend the person into God's keeping. These can be a small, quiet ceremony or a large occasion in a packed church. Everyone is entitled to either a burial service (funeral) or to have their ashes buried in their local parish churchyard by their local parish priest regardless of whether they attended church or not. If the churchyard has been closed (as is our case), then the Local Authority will provide alternative places of burial and the minister can carry out the service there instead of the church or crematorium."

Typically, contact with the Rector is made through the funeral directors to arrange a funeral service.  Thereafter, the Rector or other officiating minister will make contact with the family to arrange to visit them. To explain, plan, and pastorally support the family through and beyond the laying to rest of their loved ones in conjunction with the support of the funeral directors.