@catherinajtv the document you have linked is a 'good practice' document written by a Government department and a special interest lobby group who state various things but do not actually link them to the specific relevant legislation.
The specific mention of single sex spaces says -
'Separate and single sex services
You may only treat people with the protected characteristic of gender reassignment differently regarding access to services reserved for their self-identified gender if you can demonstrate that the way you have approached a situation is a ‘proportionate means of achieving a legitimate aim’ and so is objectively justifiable.You need to be able to
show that there is no less discriminatory way to achieve the aim.
Unlawful discrimination against people with the protected characteristic of gender
reassignment is not acceptable and consequently the exception has to be used in exceptional circumstances. Decisions made cannot be based on personal prejudice but on evidence of detriment to others, and the service provider will need to show that a less discriminatory way to achieve the objective was not available. '
There is detriment to others (women, who are also a protected characteristic) so single sex toilets are a proportionate means of achieving a legitimate aim. There is various evidence that mixed sex or 'gender neutral' toilets are detrimental to women. The English Government even agrees - <a class="break-all" href="https://web.archive.org/web/20210515234830/www.telegraph.co.uk/news/2021/05/15/exclusive-return-ladies-gents-lavatories-ministers-tell-architects/" rel="nofollow" target="_blank">web.archive.org/web/20210515234830/www.telegraph.co.uk/news/2021/05/15/exclusive-return-ladies-gents-lavatories-ministers-tell-architects/
I also suspect that the group who have 'advised' the Government are very much in the line of Stonewall's 'representing the law as they want it to be, not as it is' (as pointed out by an independent review of the University of Essex).