Who Is Legally Responsible For Making Medical Decisions For Someone Under An Anaesthetic?

The ultimate responsibility for someone who has been given an anaesthetic lies with the medical team who are performing the operation/procedure.
Prior to the operation the person will have signed a consent from which legally gives the medical team the power to perform procedures which are judged to be necessary, even if they are unforeseen.  In an emergency, if the patient cannot sign the form, then their next of kin would have signed.  So in effect a patient signs over their own and their relatives' rights for as long as they are unconscious.  

The legal and medical responsibility then becomes the responsibility of the consultant in charge.  However, if a secondary procedure is viewed as necessary, this will usually be comunicated to the patients' relatives at the earliest opportunity, to ensure that they are fully aware of what is going on.  Legally though, the relative would not be able to prevent the secondary course of action from taking place.