With regards to public attitudes to political transvestism and related issues, I’ve been trying to find this again for a while, since seeing someone post it on the twitters.
A plurality of British people (the ‘agrees’ were the single biggest group but less than 50% of the sample) agree that an individual should have the right to personally identify with a gender other than what they were born as - but that, literally, is the extent of it. The pubic disagrees with every other statement flowing out from that in terms of rights to spaces, participation, legal accommodations etc.