Your question implies that a contact order places a duty upon the child. It doesn't; it simply places a duty upon the parent, with whom the child resides, to permit contact.
At the age of 16, any young person can leave home and have no further contact with their parents, if that is their wish. Even if the young person is reported as missing, and then located by the police, the police may not disclose their location to the parents. It therefore follows that a 16-year-old can refuse to have contact with one parent. The contact order is only breached if the other parent seeks to block access.
Chris