Emily's Cat
Rarely prone to hissy-fits
Certainly it's an evolving situation in public opinion.
I did some google searching to see what I could find and what I found was mostly a reluctance to ask blunt questions. So, when asked if people supported transgender rights, surveys found that most people did. No one ever wants to say, "I am against rights." As with so many other things, the way the question is worded influences the answer. A lot of polls were split nearly evenly on bathroom questions, but I didn't see any poll that asked specifically about locker rooms or changing rooms.
If you could convince me that only a small minority of women were concerned about this sort of thing, that would probably change my attitude. Until then, I'll be supportive of the privacy rights of women and girls who do not want to be seen disrobed by male bodied individuals.
There is definitely some nuance. Here's a poll from just over a month ago, asked of Britons. If you look through the chart, you can see some clear impacts of nuance.
For example... The vast majority of people think that people should be allowed to "self-identify". But the vast majority also disagree with making it easier to legally change one's gender, or that transgender people should be allowed to legally change their gender without buy-off from a doctor and without having lived as the target gender for a good period of time. That pretty much boils down to "People can call themselves whatever they like and I'll go along with it, but if they want the legal resources of the gender they identify as they have to be medically supported and show commitment to it - lip service and a claim alone isn't sufficient."
As another example, there's pretty good agreement that transgender people should be allowed to use the changing rooms and toilets of the gender they identify as... but that support erodes pretty quickly if the transgender person in question has not had gender reassignment surgery.
Where does the British public stand on transgender rights?