This school makes me uncomfortable and I want to scale back interactions

Then you need to follow your gut reaction, but as kindly as possible because it could just be that this woman is keen to make friends for herself and her son, and is going about it the wrong way, which is not a crime, just an inconvenience for people who find it too much.
If you don't want to be friends then you need to avoid asking for favours even when desperate!

No of course you are not beholden to her, but she sounds like the type to smother her friends and maybe she struggles with boundaries. Whereas you sound like the polar opposite, a bit prickly and suspicious, with a wall up around you.

You are just a poor match as friends and she hasn't cottoned on yet. But if she's as intense and desperate to over-involve herself with you as you seem to think, then she might be a bit thick skinned and difficult to shake off. That's all I meant.

Is my reaction here too strong ?

Just going by the info in your OP, yes it's way too strong. You seem to be really obsessing over nothing much. But you can't help how you feel and if something about this woman has spooked you then you don't need anyone else's permission to back off from her. Be prepared for it to be difficult to extract yourself though. If she's the kind of person you are suggesting she might be, then she's not going to let you go without a fuss.

Funny, I'm going the other way to most PPs.

I don't feel like you're overreacting, OP. I think you have subconsciously picked up on something that is raising alarm bells for you, and that you should listen to them.

You're not obligated to keep this person in your life, and certainly not obligated to have your child be with someone you don't 100% trust.

She sounds like someone who is desperate to be seen as helpful, needed and a friend, but IMO she's crossed some lines. I completely understand why this feels suffocating. It's completely possible to be kind and helpful without being interfering and trying to be overly involved in someone else's life, which is what she's doing.

It's not normal to pretend that your DD was upset at being collected by Grandma and not her. That's very weird. And creating an imaginary scenario in which the teachers get involved is even weirder. It's as though she's pretending to be a member of your family.

She seems to be unusually keen to pick your child up and take her home. Her DS asks if he's coming home with you. All of it is very strange and I'd keep my distance too. I'm not saying she's some kind of criminal, just that your discomfort is understandable and you don't have to allow her collect your child.

Some people are missing that she's picked up your child only once, and you've picked up hers only once. It's not as though you're regularly desperate for help and turning your nose up at someone who's willing to give it.

I see the OP's perspective. This woman has overstepped the mark and lied. I would just be polite in passing from now on. Don't think the worst, she may have misinterpreted something your child said. But I'd keep my distance for a while.

then act on it.

You sound just as over the top as her to be honest op.

If you have a bad feeling about someone I think it’s a good idea to back off, be polite but not encourage them too much. She may just be an intense person or trying to be helpful but it does sound a bit OTT from what you’ve said.

Is the other mum working class whilst you are middle class OP? Because her behaviour sounds normal northern working class behaviour to me.
I have met someone twice who is helping me out with something. This is what we do. I met one of her friends who hugged me and asked my advice on a personal matter twenty minutes after I met her. We don't all keep people at arms length.

To be honest, I find the OP’s incredibly melodramatic take on it all (assuming the other mother was ‘lying’ and dashing about cross-checking the grandmother’s take on it all, and her DD’s story, as if she’s in a detective novel, and then starting an internet thread) far odder than the other mother’s behaviour, which is fairly self-evident.

She’s desperate to make a friend, and for her DD to befriend the OP’s DD, so she was delighted to be asked to pick up the OP’s child as evidence of the beginning of a bond. So she was very disappointed when the OP made other arrangements, and just projected that onto the OP’s DD because that’s what she’d love to happen, rather than the OP’s child saying ‘Oh, fine’ and going home with her grandmother.

There’s no sinister intent, she’s just one of those characters who show up on Mn all the time, terribly worried that their Reception DD doesn’t have a ‘best friend’ by Christmas, and upset they haven’t made ‘school gate friends ’.

Agree..OP sounds a little bit paranoid, especially with the ''up in my business'' that's a strange term to use, as is the ''Grown woman''...most people aren;t that touchy.

Doubt this mother cares about op's 'business' {whatever it may be}, just wants to help out {and to be helped out in return in an emergency- eg, picking up each other's children}

There are some very good points here that even I - who am 100% behind the OP in terms of her feelings - didn’t think of

I can understand your response to this - I have a similar person at school. I don't think they mean any harm and actually we have built a friendship but I have had to be quite clear in my mind where my boundaries are and develop a bit of a thick skin for all the perceived disappointment/sadness etc my children apparently experience when I get things wrong...
I wouldn't burn bridges - she sounds kind and helpful- but maybe decide how close a friendship you want and practice letting comments wash over you

There are two ways of having friends. The light acquaintance and fairly disinterested type of friendships OP you describe with other school mums. Or proper friends where you care about each other and help each other out.
Wanting to make proper friends with someone is not being intense, but if it is not what you want, then fine put a stop to it.

Don’t be silly it’s not ‘intense’, it’s perfectly reasonable rational decision on my part and I explained exactly why…

I have no problem smiling and saying hi, but I’m not there to make pals. I also feel strongly that I need to protect my vulnerable child.

Because she thought she was? And liked the idea that the OP’s DD was so keen on her DD that she was disappointed not to be going home with her?

This hasn’t convinced me at all that it’s not really weird! She’s a grown woman.
Even if she genuinely thought the daughter was upset she could just say ‘she seemed disappointed…’ Not make a big deal out of it like the OP has said she did and say she’d had a word with the daughter if the teachers mention it.

If the Mum was a Dad - would you feel the same way?

If it was the Dad all up in OP's face and taking the DD back to their house while his DS got changed without OP's knowledge until after the event - would you think that was fine and he was just 'lonely' trying to make friends?

If it was a Dad texting asking questions and offering to collect the DD without being asked, would you think it was creepy or friendly?

If a Dad had said the DD was really upset not to be coming home with me - when that was not the case at all - would you be saying the OP shouldn't be demonising him?

It's a genuine question.

Because to me, you're ignoring red flags purely because this person is a woman and a Mum. Rather than just accepting these as red flags.

I actually think you're the odd one here

I realised. After I wrote that…😀

Bloody hell, that response in itself is a bit intense.

I also hate the comparison thing but just don't engage in it? Keep it light?

Presume she just meant to write "school mum"?