To get back to the original OP, re women not being called in for smears whilst men are - this is a symptom of old legacy computer systems, and our 'faith' in computers getting it right. But computers can only do what they have been told to do.
I imagine that the patient database has been set up with a field to indicate the patient's sex (and another for name, NHS number etc.) and that the programs expect to find M or F in that field. And when a program sweeps this database it makes itself a list of all those with the F marker to call them in for a smear. (Obviously simplified, but you get the gist.) So for a male to be called in for a smear, his marker must be set to F.
Whoever decided that medical records would be set according to patient preference rather than objective reality needs to put in a proper solution, not this botch job. Sex matters - male and female bodies react differently to drugs, have different e.g. heart attack symptoms, and have different body parts (e.g. cervix/prostate). It is not in the patient's medical interest for their records to be wrong.
It could be fixed. The patient record could have a field for sex and another field for genderID. The sex could be accurate so that the body could be treated correctly, and the genderID field would indicate how they are likely to present and how they would like to be treated.
It is difficult to make such a fundemental change to computer systems. I remember the amount of work that went into the Millennium Bug, changing all the database date fields from DDMMYY to DDMMCCYY and rewriting all the programs' date handling; but it had to be done. This is the same.
The original data structure does not support the current data requirements - it needs to be expanded. Recording a male as female and vice-versa was and is a 'quick and dirty' fix to meet an immediate (perceived) requirement, but it needs to be revisited and done right. This will be costly for the NHS, who seem to have a sprawling unintegrated IT environment - inevitable given the sprawling unintegrated nature of the NHS, perhaps. But until they do, females will be invited for prostate exams and males will be invited for cervical smears, and health needs specific to their trans status will be missed.
This is down to the misplaced 'faith' in computers - the decision makers thought the IT departments could wave their magic wand and make it work - but as I said, computers need to be told what to do, and that takes work that has not happened.