AIBU to think this woman is rude and obnoxious?

I don't see anything rude in not wanting to live near chavs- who would??

But she sounds like she thinks she better than everyone (and that her daughter is, too!) I couldn't be arsed with that so would distance myself.

Anyone who expresses to a near stranger that they “don’t want to live near chavs” has something to prove. Its grossly lacking in class or a basic ability to read a room.

The examples you have given are a on the fence for me. Her delivery is cringe but people do use subtle ways to discover the 'nicer' parts of Town. Phrases like 'Suitable for families', 'great community spirit' etc., ring any bells. And who doesn't want their child to go to the best school?

Your friend sees something in her but it doesn't mean you have to. Does she have a sense of humour? Can you turn her rudeness/bluntness back on her?

'Are there any chavs in the area?' Reply 'Not until you moved in'.

'My daughter is so far ahead of ordinary children' Reply 'There are schools that cater abnormally tall children, that's amazing'

She could learn to laugh at herself, rein it in or she will distance herself. You hold the power. She is not at your school and she doesn't really know what is going on in your life. I would mostly ignore her texts and only reply to the odd text with things like 'So busy, hope all is good' 'Lots going on here, hope house hunting is going well'. Reply with statements, not questions. That stops it being awkward when you do run into each other.

Good luck OP

she is using you and will drop you when you are no use to her so i wouldn't have any qualms about dropping her first. She is the one moving to a new area and needing to make friends and connections, not you. Stop being so nice and stop helping out this nasty person.

She sounds like real jumped up pretentious little prat. Who has come up from nothing. Unfortunately, these types are the worst.

She doesn't seem to have much to offer.
Just ignore her.

Maybe try and call her out on her bullshit in a way that puts it back on her. Make her feel uncomfortable with you and, with any luck, she won't try and contact you again.

Eg. 'You keep mentioning chavs. Do you think I'm a chav' add intimidating stare
or even better
'You know no one uses 'chav' any more? It makes someone sound very old and judgemental....'
Or similar

She is a common denominator person. She has lived in lots of different places and not liked any of them. In addition to this, she has fallen out with almost everyone she has ever met yet spectacularly fails to see the common denominator.

Part of me would want to stay in touch with her just to marvel and laugh at her nonsense, she sounds like a comedy character from the Fast Show. I would be polite but not give her any means of gaining leverage into my life.

I’m intrigued as to how the mutual friend handles her (other than palming her off onto yourself..)

She's, as you say your friends friend..

Leave it at that and block her number, if you see her out just keep walking, that doesn't make you rude it's makes you someone that won't put up with her treating you rudely.

She sounds like hard work!! (Please read "horrific").
Youre not under any obligation to be her friend. You've been kind and welcoming and she seems very socially unaware, but that's not your problem.
If you cross paths it might be awkward but unless she asks, again, you're under no obligation to explain why you stepped back. Unless you want to!

Is this one of those vanishingly rare occasions where you can get the permit to deploy the Mumsnet head tilt, how rude, tinkly laugh sequence?

I once knew a mother on the school run like this. I think it's just a (crap) personality type. According to her, she was better than everyone else and she wanted everyone to know it, too. Frankly, I'm amazed that your woman has got any friends, the one I knew certainly put everyone off.

She sounds awful. I wouldn't give her the time of day, personally.

lived in several places but liked it nowhere.

I had a male friend like this growing up. Not rude like you describe but constantly complaining about there they lived such and such a place would be so much better, grass always greener somewhere else. Unfortunately every time they moved to where they perceived as better it wasn't and they moved on again. (I am talking different countries too). I eventually suggested that maybe they are not happy and no amount of moving would solve that.

She sounds ghastly. I wouldn't answer her texts and treat her with the derision she deserves.

I would have looked confused and said "What's a chav?" (just to hear what her stupid response would be).
About the school I would have said "Oh what school is she going too then?" and then done that "Mmmm interesting..." thing while nodding my head (to show I didn't really care that much).

I would be polite due to mutual friend so I would reply occasionally. But definitely wouldn't be spending any time with her.