The parental right to determine whether or not a child below the age of 16 will or will not have medical treatment terminates if and when the child achieves sufficient understanding and intelligence to enable him to understand fully what is proposed. Note that 'Gilliick competence' relates to the particular child and the particular treatment, and there have been cases where a 17-year-old has been found insufficiently competent to refuse medical treatment, while in other cases much younger children have been deemed sufficiently competent. In addition, where a child is 16 or 17 either parent or child can consent to treatment independently (though neither can override the other or exercise a veto). The court can, however, override the wishes of both where treatment is vital to the child's welfare.
The Access to Health Records Act 1990 further complicates the picture in allowing a child under 16, deemed 'Gillick competent' by a doctor, to veto the parent's access to medical information held by that doctor, even though the parent can consent to treatment which the child cannot veto.
It is for the doctor to decide whether or not an individual child is Gillick competent or not. The courts are generally reluctant to do so.
See also: this Wikipedia article.