YouGov survey from June 2020 - I find that a useful graphic.
yougov.co.uk/topics/politics/articles-reports/2020/07/16/where-does-british-public-stand-transgender-rights
As mentioned previously, sports is the only area where attitudes have noticeably changed since the 2018 study. The proportion of people opposed to transgender women taking part in women’s sports rising by seven points, as well as a five-point increase when it comes to transgender men.
Every single social group is now as or less likely than they were in late 2018 to back transgender access to gendered sporting contests. This is especially the case when it comes to transgender women.
There is also a 2022 survey:
yougov.co.uk/topics/society/articles-reports/2022/07/20/where-does-british-public-stand-transgender-rights
Access to sport: Britons oppose transgender athletes taking part in sporting events for their new gender
There has been a recent spate of sporting bodies imposing bans on trans athletes competing under their new gender, including British Triathlon, the international swimming body Fina, and International Rugby League in the last month alone.
The trans sport debate has focussed primarily on the ability for transgender women to compete, with opponents arguing they have an inherent physiological advantage even after transition.
The public are strongly opposed to trans women athletes participating in women’s sporting events, by 61% versus just 16% who are supportive.
It is not clear, however, how much of this is represents acceptance of the argument that trans women athletes have an unfair advantage, because Britons also tend to be opposed to transgender men competing in men’s events. Such opposition is lower, however, at 48%, with 29% in favour.
So 61% of people are now against transwomen in women’s sport.
Lia Thomas and Emily Bridges have done sterling work.