The way the law currently stands in England is this.
It is not illegal for a man to go into the Ladies.
It would normally be illegal for the owner of the building to stop a man doing something that a woman can do, eg go into a certain set of toilets, because that would be discrimination under the Equality Act 2010.
But in certain circumstances, such as toilets/changing rooms etc, it is permitted for the owner of the building to allow women in and keep men out, because they are allowed to have single-sex services in these limited circumstances (Equality Act Schedule 3).
As a result, women currently can feel confident to challenge a man who comes into the Ladies because the owner of the building appears to have provided a single-sex service, and men should not be coming in.
The position gets clouded by three things.
Firstly, it is illegal to discriminate against a transgender person on the grounds of their transgender characteristic This is often used by TRA to say that transgender people must be able to access any areas they want, otherwise it is discrimination. This was the argument being used in the BBC piece that the OP linked to. But this is wrong. Discrimination happens when you treat a person worse than you would do if they did not have that characteristic. So in our present example, a man who is transgender, who is being stopped from entering the Ladies, is not being stopped due to his transgender status, it is because he is a man. He is being treated the same as a man who is not transgender. This is permitted due to single-sex services, as above. It is not discrimination and it is not against the law.
Secondly, a person who has obtained a Gender Rights Certificate must be treated for all purposes as if he/she was the opposite sex. What does this mean for the single-sex services? It has not been tested in law. On the face of it, it means that anybody with a GRC can go into single-sex services designed for the opposite sex and nobody can stop them. This is why self-ID is such a big deal, because it means that anyone can get a GRC.
Thirdly, real life. In practice, nobody asks for documentation before going into toilets. Anybody can walk in. But if a person is obviously a man, at the moment women can shout and complain and challenge them. If self-ID is introduced, those women who challenge and complain may find themselves being sued, being arrested, or at the very least, being chucked out of the venue, or being told that they are transphobic and must accept this male person.
The big problem is that so many large organisations have fallen for the lies about what the law currently says, and have changed their policies to say that self identified gender is what matters.