This bit sums up their position on single sex spaces. I really wish this could be tested in court properly because I don't believe it. I think they're wrong.
They say the bar for excluding tw from female spaces is very high.
There has to be a very specific set of reasons to exclude trans women from single-sex spaces. Although we dearly wish that no single-sex spaces wished to exclude trans women, trans men or non-binary people, we also recognise that, for the minority of spaces that want to, it is probably not a particularly safe service for those trans people to access.
So their reason for it being legal to exclude tw is not because women need that sometimes but because women who want to do that are nasty bigots who will make tw unsafe anyway.