What would the impact be if gender identity were the protected characteristic?

It would not protect women in the way that women are protected now

There would be no single sex exceptions - so single sex jails , changing rooms , sports , no monitoring of wages or health by sex because they are all discrimination based on sex which is only allowed because sex is a protected characteristic

These would also not exist for gender categories as these discriminations are only allowed for sex because of clear , objective , measurable reasons

You would need to prove the need for single gender sports for example and if a significant proportion of women don't identify as such you won't get the clear differences needed to establish gender categories for anything

Also belief (and lack of belief) is already a PC. GC is already protected in law following the MF case.

*Wouldn't this

But in fact, if they are trying to argue everyone has a gender identity, surely it would also refer to people who are not trans, and the whole range of beliefs that come with that, which would include being GC?

Wouldn't lead to people having to pretend (lie) to have the PC of gender identity to be able to be protected. You are right that It may not work out how trans activists want, my guess is that it'll make things worse for everyone. As PP pointed out the case of the fake breast teacher in Canada (possibly an example of where people are starting to game the system). On the other hand cases like Jessica Yanniv (and many other cases) are a good example of the damage done to multiple women by these laws.

The law should be clear and specific and predictable. There is nothing clear or specific about the term "gender identity". What does it even mean, how is it balanced against belief or sex? Nobody knows.

Oh god.

Ten years ago it would have sounded equally implausible that a sex offender would be put in a women’s prison because he says he’s a woman….

It would become The Protected Characteristic. TRAs are trying it now.
They are dismissing religious beliefs in personal modesty as irrelevant.
They are trying to conflate sex and genders. They are trying to say a bloke who puts a dress on for a [fill in your own word] is the same as a person who is considering getting drugged and mutilated to match the idea in their head. #NotAMentalIllness
Make it a protected characteristic and it will only get worse.

That's an interesting question. The problem with gender identity is that it is so ill-defined by nature. There would be long ranging consequences far beyond women's rights IMO.
"I am X because I feel like X" means that pretty much anything could be claimed to be a protected characteristic.
Take those adult males who claim to be six year old girls. If the entirety of that is claimed as a gender identity, (which it could be as per the current fuzzy definition) then legally they might claim the right to go to primary school.
That might seem an extreme example, but that's the risk when you put something indefinable into law as a 'right'. People will take advantage of it.
It'd be completely unworkable unless a non-circular definition of gender identity was included.
However, the problem with gender identity is that it is all based on stereotypes so either you have an undefined protected characteristic that anyone can take advantage of or you are protecting the performance of gender stereotypes within law which would have it's own negative consequences for women and girls (that Canadian teacher with the falsies for e.g.)
I think it's one of those things where it sounds like a simple answer on the surface but it'd actually have quite serious unintended consequences.

I don’t see why there would be protection for those who claim not to have a GI. If you applied that thinking to the PC of disability that would mean the able bodied were protected. Lack of belief is already protected within the Belief PC. Having GI as a PC would be a problem for women. How to balance the PC of sex with GR is already a headache. Adding GI into the mix (which, lets face it, has no objective meaning) could create new issues.