Well what a depressing rabbit hole this has turned out to be.
So, yes, the Higher Education Statistics Agency does indeed demand a huge amount of data on 'characteristics' - most of which are not EA2010 Protected Characteristics, and some, such as 'Gender Identity' are obviously controversial at best, yet this government agency takes the existence of gender identity as an absolutely given fact.
Here is the list of what Higher Education Institutions (so Universities, basically) must send them every year.
https://www.hesa.ac.uk/collection/c22025/index
If an institution does not do this, they can be prevented from taking on students etc. So this nonsense is coming from government agencies (albeit eagerly followed by HR departments)
Interesting that 'Gender Identity', 'Transgender status', 'Sexual identification' are all listed (but NB the ridiculous framing: sexual identification says 'This field records the sex of the member of staff', so why 'identification'? That's right: because you have to put 'identification' in all these things otherwise gender ideology collapses under its own ridiculousness).
But look: the best practice recommendation is that the 'Transgender' question is asked every year.
But the trans question should be asked alongside the 'sex' question. So this rather raises the question of why my employer does not ask about 'sex' at all.
Preferred pronouns are, of course, nowhere to be found on this list. What on earth could they tell anyone? How can you tell the female "she/her"s from the male "she/her"s?
So yes, I will try to take this further. Showed some of this to a colleague who has not previously paid any of this any real attention, and he was perplexed with lots of pulling faces and head scratching before saying that something seems wrong here and he was going to start finding out about it too.