I think this is the most problematic part of the judgment but I can see the logic.
Your starting point needs to be, "Who is this space or service actually for?"
So in the context of a single sex space, your female changing room is for female people who need to get changed without male people present.
You have your group sharing a protected characteristic. They are people who share the protected characteristic of sex, i.e. in this case they are all female.
Then you need to apply the law to figure out whether this situation meets the criteria to be able to apply the exemption in the Equality Act. There probably hasn't been a lot of analysis about which of the criteria we are applying to justify single sex changing rooms, because society as a whole generally accepts the legitimacy of single sex changing rooms. But let's say you are relying on the justification that the users of that space might reasonably object to the presence of the opposite sex. Is it reasonable for women to object to the presence of the opposite sex when they are getting changed? Clearly, yes, as a society we believe that that is reasonable.
Then you also need to be able to demonstrate that this is a proportionate means of achieving a legitimate aim. The legitimate aim is to ensure women's safety and dignity, and it is proportionate because men are also provided for. (You can also refer to the justification that you are providing the same space to both sexes and that it is more effective to do so on an equal but separate basis.)
So if this space is for female people, to ensure their safety and dignity when they are getting changed by having no male people present, all your analysis needs to be focused on that group of people and the impact on them.
The mistake many people make is approaching it from the point of view of a trans person who is being excluded. But the space is not for trans people, it is for female people, and so you need to approach it from the point of view of those female people.
So, does the space serve its purpose from women's point of view if men are allowed in? No, clearly not. The purpose would be completely defeated.
Does the space serve its purpose from women's point of view if trans women are allowed in? No, for several reasons. Firstly, there is no way to distinguish between a genuine trans woman and a man who is just claiming to be a trans woman in order to access a women only space. If you are letting trans women in then you have no ability to exclude a man who decides to call himself a trans woman purely for the purposes of accessing that space. Secondly, even if there were a way to distinguish between a genuine trans woman and a man who is just pretending to be one, the women in that space are still likely to perceive the trans woman as a man. They will still be able to see that the trans woman has a male body. They will still be able to see that the trans woman can physically overpower them. Many of them will still be uncomfortable with the presence of a trans woman in the space. None of them will have any way of knowing whether the trans woman still has a penis or whether they have a gender recognition certificate. And an individual trans woman does not represent any less of a statistical risk to women than an individual man does. So the space is no longer serving its purpose as a female only space, and the use of the exemption can no longer be justified because it is no longer actually achieving the legitimate aim.
But what about trans men?
Once again, you have to think about it from the perspective of the female users of the space.
The law is clear that if a trans man does not pass, they should be able to use a female only space. No woman is likely to be intimidated by finding someone like Elliot Page in the women's toilets. We can tell they are female even if they have no breasts and a little beard. But if a trans man does pass, to the extent where the female users of that space cannot tell that he is female, the impact on them of him being in that space is the same as if a regular man were in there. Thus, the space is no longer serving its purpose or achieving the legitimate aim.
Where the judgment falls down is that it says that in these circumstances a passing trans person (who will almost always be a trans man than a trans woman because the effects of testosterone on the body are irreversible and so trans men stand a much better chance of passing than trans women do) should not use a single sex space for either their own or the opposite sex, but they should not be in a position where there is no space for them to use. It does not elaborate on what the passing trans person is supposed to do if no unisex space is available.
In reality I think this is the situation most passing trans men are already in. The trans men who have posted on here have said that they tend to avoid single sex spaces altogether and seek out unisex spaces or use accessible ones where necessary.
But I do not like the principle that a small group of people is uncatered for, even if it is due to their own decision to drastically change their physical appearance.