I think, given the wording in the manifesto and what Pidcock just said, I think they will have to (at least) keep the EA exemptions as they are.
This has been discussed a bit on the "Labour promises to keep single-sex exemptions" thread.
This is what the manifesto says.
Ensure that the single-sex-based exemptions contained in the Equality Act 2010 are understood and fully enforced in service provision.
Single-sex exemptions are about different or separate services for men and women. They are nothing to do with transgender people. Those come under the exemptions for gender reassignment discrimination. Labour haven't mentioned those.
See Sections 26-28 of this part of the EA2010.
www.legislation.gov.uk/ukpga/2010/15/schedule/3
Yes, they might keep the single-sex exemptions as they are, but they want to redefine sex by opening the GRA to self-ID. So there will be single sex spaces, but they will be for people who share a legal sex, who might in fact be either male or female.
If this is not what Labour intends to do, they need to be clearer about their intentions.