Surely it's only discrimination if they claim to be providing a single sex service?
Saying men can't join is sex discrimination, full stop. Regardless of who they are.
The question is whether they're legally permitted to discriminate like that.
Why though? I.e. a Veteran's men's org is only for men who have been
Veterans. Why would it be different?
The "veteran's" bit is not relevant to discrimination legislation, as being a veteran or not is not a protected characteristic. Nothing bars you from discrimininating against non-veterans (or veterans).
But it's sex discrimination against women. Which is permitted if it's to provide a single-sex association. That's the single-sex exemption.
Being single-sex is in many cases deemed permissible with a low bar, and for it to be possible, then obviously you have to be able to discriminate against the other sex.
A "veteran (men or women)'s organisation" is also fine - no sex discrimination at all.
A "(veteran men's) or (women's) organisation" would not be permitted because you were requiring men to be veterans, but not women. There's no reason for such sex discrimination - you're open to both sexes, so why treat them differently?
They can't exclude any men if they are not a single sex association.
To clarify - they can't exclude men because they're male. They have to be excluding them on a sex-independent basis, for the same reasons they'd exclude women.
But if they're open to all women, then yes, they'd have to be open to all men.