Enby they/them pronouns - what's the law on this?

Is the Dr (singular, no sex specified) in today?
No they aren’t.
Still not odd.

In practice, your sentences would read
They are going to the park.
This is Jane, they are the manager.

I think a citation is probably required for this.

If my name is Caroline but I ask people to call me Caz and they instead keep calling me Caroline, that's not harassment.

Non-binary does not exist in law, so there's no difference.

Really try these
She is going to the park
They is going to the park.
This is Jane she is the manager
This is Jane they is the manager
Only 1 of each set of statements is grammatically correct the other isn't. They is a plural and shouldn't be used yo refer to a single person

Neither of those ‘sound odd’.

Using ‘they’ in place of she or he is grammatically correct,

Is it though?

If someone says, 'has anyone seen Anna?' It sounds odd to say 'they went that way'. Or, asking 'did Maria want to join us for our lunch walk?' doesn't expect 'they didn't say' as a reply.

You are incorrect about the grammatical use of they.
However it can be harassment to continuously use tge correct sex pronouns for someone who has asked you not too but that doesn't mean you have to use the chosen pronouns avoiding using any pronouns is a legally acceptable compromise.
Also.if this nb person complains about lack of pronoun use and/or corrects people repeatedly that would be harassment and belief discrimination by her to her colleagues

We all have protections under human rights legislation from being forced to tell lies.

No you don't you just use her name or you when talking to her.
If she starts demanding they them or having a go at people for just avoiding pronouns it will be time to have a word with her about her inappropriate behaviour and that she cant demand people use her chosen language all she can do is ask not to be referred to as she/her.

Using ‘they’ in place of she or he is grammatically correct...

..when you don't know their identity or sex.

When these are known, it is grammatically correct to use the customary sexed pronoun. A colleague's identity and sex will be obvious. The OP already knows that the person in question is female.

Coercing people into pretzelling their speech with grammatically incorrect pronouns is the nearest thing to harassment in this situation.

Intentionally not using someone’s correct pronouns can be seen as a form of harassment in UK workplaces. Using ‘they’ in place of she or he is grammatically correct, so calling someone ‘they’ rather than ‘she’s is completely acceptable, but if you’ve stated you don’t want to be called ‘they’, you want to be ‘she’ and a person keeps calling you ‘they’, it could be seen as harassment (and the other way around).

But, is this ever going to happen? Personally I hate being called she, I prefer people to use my name when I’m in their company as I think it’s very rude. When I’m not there, they can call me she/he/they/banana - I don’t really care. I usually refer to people I don’t know the name of as ‘they’ anyway. I don’t think you need to make this into a drama preemptively, it might never be an issue.

I don't see what grounds there would be for a complaint of discrimination.

Discrimination claims are based in the protected characteristics in the Equality Act 2010, one of which is "gender reassignment". Gender reassignment in the GRA is male to female or female to male. The notes on applying for a GRC on gov.uk specifically underline this: "You can only apply to be recognised as male or female. Non-binary genders are not legally recognised in the UK."
https://www.gov.uk/apply-gender-recognition-certificate

And it has been established in the Ryan Castellucci case that the GRA does not allow for the legal recognition of non-binary gender.

Is it just the email signature the company don’t want changed? I don’t think there is necessarily a law concerning email signatures, but it could be seen as discrimination/harassment.

Turn it the other way round and ask in what way this employee could raise a grievance.

If the email signature format rules apply to everyone, without exception, then I'm not sure what grounds she would have here.