It is not illegal to miss a doctors appointment.
However if the child was under 16 and needed to see a doctor urgently (i.e for an chest infection or something that requires immediate treatment, as opposed to something like acne or ingrowing toenail's), there would probably be some sort of parenting offense involved by not taking the child to the doctor (unless it was just forgetfulness that meant the appointment was missed). If the child is over 16, it's no longer the parent's responsibility to take the child to doctors' appointments (unless the child is not mentally capable of looking after themselves).
If the appointment was to discuss referral to another specialist, or some non-life threatening condition (i.e a non-urgent appointment), it is not "illegal" to not take the child, but it is not helpful either.
Obviously if the appointment was booked knowing that the child would be with the other parent, the other parent should have been told as soon as possible about the appointment to check they were available to take the child (and if not, rearrange for a convenient time). And you should always notify your doctors surgery if you are going to miss your appointment.
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