Depends entirely on the language of the law, not what right wing people think. The UK isn't like the USA in this regard, or at least Scotland isn't ...yet.

That's part of what their protest is getting at — as you highlight, even "biological sex" is pretty complex (In science, I have heard that the "three G's" (Gonads, genetics, genitals) model is the standard definition, but scientists who research biological sex seem to consider this an extreme oversimplification). Fuzzy definitions like this are fine in science, but things get much messier when we try to write these things into law. One of my problems with the recent Supreme Court ruling on transgender rights is how they use the phrase "biological woman", as if it is a simple matter.

I find this especially striking because I'm a cis woman who has plenty of experience of being treated poorly due to being a woman, and I feel like my "biological sex" (as in gonads, genetics and genitals) don't factor into it much; far more significant is whether I am perceived as a woman, and this is why "gender" can be far more useful than "biological sex" in these discussions.