Arguments against strengthening the rights of transgender people, on both right and left, are often posed through a false counterposition with women’s rights. Those making them frequently accuse trans rights advocates of ignoring “women’s views”, “women’s concerns” and so on.
Men in the UK agree that “a person should... be able to self-identify as a gender different to the one they were born in” 43% to 33% (the rest say they don’t know). Women agree 57-21.
As you’d expect, younger people are much more strongly in favour. 18-24 year olds support self-ID 64-14. Over-65s only narrowly support it 35-34.
Labour supporters support self-ID 70-13. Tory supporters oppose 35-42. Remainers support 62-19, while Leavers oppose 39-41.
Men agree that “schoolchildren should... be taught about gender identity in schools” 44-38. Women agree 52-28. Men agree that children should be taught about "gay relationships" 53-29, women 61-22. For Remainers these figures are 65-20 on teaching about gender identity and 74-14 on teaching about gay relationships; for Leavers 34-47 and 45-37.
Most of these figures are relatively encouraging in the context of major attempts to generate a backlash against trans rights.
A fairly large qualification is that in the last year the proportion supporting self-identification has fallen from 56 to 50% and the number opposed risen from 23 to 27%. That could be a margin of error issue; it could also reflect the impact of reactionary campaigning.