It's not really a political left/right wing divide. Up till a few years ago, it was virtually impossible to see any difference in left and right trans policies, in the US or here. It's possible to get to twaw and twanw from both a left and right wing perspective.
I think its a 'believe authority' thing. If people believe that the government laws are generally sound, they have had work base training, or trust stonewall or the bbc in their trans people are the most marginalised narrative, they are going to believe TWAW without giving it a second thought, or even looking into it.
Its only when really barmy things occur that people have any reason to question. And even then, they rationalise it by saying, well of course this person shouldnt be in the women prison, but tw have to use the womens toilets to keep them safe. They trust government/employers/charities to write policies that make sense, and assume its an unintentionally loophole rather than a bonkers law in the first place.
Some people are so loyal to a party it effects their outlook, but most people arent, and they are just more trusting of authority, until they are affected by it.
If the government were to repeal the GRA, and trans inclusive talks were stopped in the workplace, i doubt most people would notice or take to the streets. Because they trust the government to have made the correct decision.