Short answer: Yes, the marker on GP records is changed on patient request.
Long answer: When the new NHS digital systems were set up there was very extensive discussion about how to record sex and gender in clear and unambiguous ways that met both medical needs and patient preferences. They came up with a 2-box system to record both sex and gender, with guidance that the former was essential and the second optional.
Various people have done subject access requests on their own records and found the sex field is blank, and the gender one has been filled in without them being asked.