Apoologies. When I wrote that I had overlooked the fact that WI currently admit trans women. However...
If you run an association with fewer than 25 members you are safe as you are not subject to the Equality Act. You can set whatever rules you want around membership.
If you have 25 or more members you are subject to the Equality Act. However, you can rely on the single sex exception if you admit only women.
If you admit women and some men, you cannot rely on the single sex exception, but that doesn't necessarily mean you are discriminating illegally. The question is more complex.
If you are applying different rules for men and women, that is sex discrimination. So, for example, the group suggested in an earlier post that admits all women but only red-headed men would almost certainly lose if a man made a sex discrimination claim against them.
If you are applying the same rules for men and women, it may still be indirect discrimination if the rules are likely to allow most of women to join but prohibit most men (or vice versa). You can still get past this provided the rules are objectively justified, i.e. they are a proportionate means of achieving a legitimate aim (and, of course, provided they don't bring another protected characteristic into play). What classes as a proportionate means or a legitimate aim is not defined in the legislation and is approached on a case-by-case basis. It depends, in part, on the effects of the discrimination. So, for example, if your rules only exclude men who are unlikely to want to join, the bar for being accepted as objectively justified is much lower than if they exclude 90% of those who would want to join.
In the WI case, since the only qualification for a man to join is that they are willing to say they identify as a woman, and WI don't ask any further questions or investigate to check that the person in question genuinely identifies as a woman, I think WI could argue that, as there is no actual requirement to be trans, any man can join therefore there is no discrimination. It is also possible that the courts would conclude that their rules are a proportionate means of achieving a legitimate aim, given that most men are unlikely to want to join the WI.
I may, of course, be wrong. Unless a man brings a sex discrimination case against the WI we will never know. However, my view is that WI might have a problem if they required a man to prove they were a trans woman before admitting them as that could bring the protected characteristic of gender reassignment into play but, as they simply take the man's word for it, I think their current position is defendable.