Gamer I suggest YOU give the GRA a good read and when you've done that you give the equality act a good read 
GRA makes it possible to legally change sex after 2 years living as a woman + diagnosis of GD. Lily does not have a GRC and is therefore legally male. Lily has claimed to have a GRC but the timeline doesn't add up. Lily hasn't been living as a woman for long enough.
The EA says that 'women' are people of the female sex. That is actual women (old fashioned cunty sort) and transwomen with a GRC (whose legal sex is female). So not Lily. Lily is protected from discrimination under the EA because of the protected characteristic of 'gender reassignment', however Lily does not share the protected characteristic 'sex' with women.
The EA then has additional sex-based exceptions for single sex services, occupations etc. to be limited purely to old-fashioned cunty women and to exclude transwomen even where they have a GRC, as a proportionate means of achieving a legitimate aim. Labour could if they wanted use that exception for 'women's officer' posts but they have chosen not to. It's not clear whether it's legal for them to have a post of 'women's officer' advertised as open to women only and then accept applications from self-ID'd transwomen with no GRC. It's almost certainly not legal for them to have all women shortlists that are open to self-ID'd transwomen as they are currently doing. AWS are a specifically named exception in the EA and are only legal based on the protected characteristic of sex.