The Equality Act 2010 does, of course also cover toilets. That's via the single-sex exemptions in Schedule 3, Part 7 (titled SEPARATE, SINGLE AND CONCESSIONARY SERVICES ETC), paragraph 26, sub-paragraph 6:
(6)The condition is that—
(a)the service is provided for, or is likely to be used by, two or more persons at the same time, and
(b)the circumstances are such that a person of one sex might reasonably object to the presence of a person of the opposite sex.
Link here
The need for separate toilets for both sexes is also explicitly stated in the relevant Building Regulations for Northern Ireland, Scotland, England and Wales. These regulations also specify how unisex toilets (single-entry facilities for use by both sexes, but only one after the other) have to be built.
A person like this complainant has no legal right to use a facility provided by the owners or operators of the building you are in (assuming this person remains legally male, i.e. has no Gender Recognition Certificate).
And any provider of such a facility would do well to remember that excluding males from a service or facility provided for females is not unlawful discrimination on the basis of sex if and only if ALL males or all males not in a position of a GRC are excluded.
As soon as you let some males who remain legally male in, but keep other males who remain legally male out, you are unlawfully discriminating against the latter. The men so excluded could sue for sex discrimination and would likely win such a case.
It's easier to do this by invoking a strict single-sex exemption which also keeps males who do have a GRC out, but providers do not know the law well enough to know this is perfectly legal and, of course, they do not know enough about how to invoke the single-sex exemptions.
We do not check people's right to use such facilities, because we have had such a strict social convention of staying out of single-sex facilities provided for the opposite sex. This taboo has been so strong and existed for so long - since these facilities were first provided for the benefit of women - that there was never any law needed that spelled this out.
No decent man would have ever dreamed of denying women the right to toilets dedicated to their own sex. But given that it's not the decent men currently in the ascendancy, we might need that law now though.