It appears to be even more complicated than that. The exemptions apply to services but not to single sex associations or schools.
From the Policy Exchange document
"Section 101 of the Equality Act sets out provisions for membership within associations. Section 101(1) prohibits discrimination in the admission or terms of admission of members. Schedule 16 sets out certain exceptions to this prohibition of discrimination. Most notably, sch.16 para.1(1) states that:
An association does not contravene section 101(1) by
restricting membership to persons who share a protected characteristic.
This means that single-sex associations are free to discriminate on the
basis of sex but not on the basis of any other protected characteristic,
including gender reassignment. Drawing upon the analysis set out above,
if Lady Haldane is correct, we can conclude that female-only associations
can exclude all legal males, be they biological males or GRC holding
biological females, subject to justification texts. But they cannot exclude a
GRC holding biological male."
So a public swimming pool could restrict it's changing rooms to biological sex if it was proportionate etc.
If I set up the MN Women Only Swimming Club and there were more than 25 members. The club would not be able to exclude a GRC holding male from membership. So if we hired the same swimming pool we wouldn't be able to prevent a biological male with a GRC from joining us but we could exclude a transman with a GRC"
Very logical...