How would you react to this 10 year old?

As a volunteer who does not appear to have much experience of managing behaviour.

What support are you offered by the school to so that you understand the school's policies and 'how they do things around here' ?

‘Go see your teacher over there’

Was your dc the only child with a parent there that was coming to them to get cuddles etc?
If so I’d maybe assume it was a bit of jealousy.
In any case I’d have pointed her towards a staff member each time

You sound like you don't have experience with a range of kids and needs tbh. I don't see much of that as rude just communication. Faces being pulled id just ignore! Just say let's go find <teacher> so she/he can make sure you're all sorted for the activity.

If teaching staff were on hand, I personally would have had a quiet word with one. I work in a First school (Foundation - Year 4) in a support role, and wouldn't put up with this type of behaviour. No staff member, would expect you to accept this behaviour either. If you volunteer again, ask what your boundaries are with the children - can you tell them off for poor behaviour or should you refer any bad behaviour to a staff member etc. You should absolutely in my opinion be able to call a child out for being rude towards you, but all schools work differently.

She sounds quite rude, but maybe she has some kind of medical/emotional difficulty. If there were any staff there, I'd have directed her towards them and said 'if you need anything please speak to miss x, but you should sit down quietly now as the film is starting please.'
If it wasn't picked up by staff you were being somewhat overwhelmed by the child, maybe just speak to them about her behaviour in a factual, non judgemental way.
By all means refuse to volunteer in future though!

"Sit down and be quiet" ??

Option
A) If you ask me politely then I will.
B) Go and ask your teacher

Well you wouldn’t shout at them, but you would say calmly that you only help polite children.

If she was being that rude then one of the teachers should have stepped in and told her to be polite and sit down. Sounds like she saw you as an easy target. Be polite, professional, but firm going forward.

If it's still bothering you I'd have a word with her teachers and see if she's always like that. I suspect if you'd been a bit firmer with her she'd have left you alone

Yes I'd have said this too. I used to volunteer a lot in primary school and I was there just for extra supervision and numbers. Any bad behaviour was always dealt with by the teachers who knew the children best.

You're within your rights to (nicely, calmly) tell her off. When I've volunteered on school trips before, the teacher said "any problems/behaviour issues etc, grab a teacher, but if it's just low level stuff and you feel you can deal with it then feel free to get on with it 😁"

Equally it would've been fine to redirect her to the teacher, or if she wouldn't be redirected just go speak to a teacher yourself - even if it meant walking over to the teacher whilst the child followed and saying something like "I'm having some issues with x..."

”sorry, I’m not a teacher, but let’s go find someone who can help you” and I’ve had led her to a teacher who could help. That makes it clear to the child that you’re not the person, and also brings it to the attention of a teacher. .

Maybe she's neuro divergent.

Hmm, interesting it started after her seeing you cuddling DD. My first thought is that she was jealous. Maybe she doesn't get much of that at home. Either way, I'd be directing to teaching staff.

Something like this id have said 'i'm not sure I want to look after this for you if you are so worried about it, you need to look after it yourself'. And handed it back.
And demands /unpleasant tone of voice would have been met with 'do you want to try asking that again in a nicer way, im afraid that wasn't a very polite way of speaking to me was it?' said pointedly.
You can get the message across without shouting etc, you do not have to be a pushover!!!

She's a natural bully and instinctively homed in on you.

I don’t have any advice OP but I do have a cute semi related story to share.

We had a girl like this at the school I used to work in when I was 18 (I was a TA in Yr6…). She was called Jessica. Tried to boss around all the adults thinking she was in charge. Always demanding things and would never be patient or just sit down quietly. She used to scare me a bit.

One day some graffiti emerged calling her Bossica. I couldn’t help but laugh (the graffiti was formally dealt with by teachers)

Anyway, Bossica I mean Jessica dialled down her bossiness after this and grew up to be the most loveliest of woman.

I know her as a close friend now as we’re in the same hobby club. Our age gap isn’t that big at all! It’s quite funny looking back! 🤣

I sometimes still call her Bossica after a few gins.

Children are very good at weighing people up and can see things in people that adults can’t.

she sensed you were an easy target and uncomfortable in the situation you were in.

you need to be firm with her otherwise she’ll become a pain in the arse for you

If this is being supervised by staff as well as parents then I would have referred her to her teacher.
She sounds as if she may have some additional needs… she may just be a bit of a pain 🤷🏼‍♀️.

Perhaps she was 'chancing ' it with you because you were an unknown face? None of her behaviour was acceptable. In future, if you don't want to get into confrontation with her, get up and speak to one of the teachers or the organisers and let them know this girl is behaving badly. It may be that she has behavioural issues and the staff can deal with her

"Go and ask a teacher" would be my response to any demands. They will know how best to deal with the child. Volunteering is supervising the reasonable kids.

Told her to go away, I don't do things for rude people. At that age, there's no excuse.