My BIL does this. I don't know about this woman OP, but in his case it's passive aggressive - I hate to use that phrase - but it really is, and there's not too much passivity about it either.
He'll turn up really early because it suits him and then sulk and make smartarse comments about having to wait when the people he's supposed to be collecting turn up, even though they're not late and we all know what school closing time is or when the train arrives and no one asked him to arrive 45 minutes early.
It's blatantly obvious that he wants to get out of giving lifts and thinks by doing that, he will or make your life a misery - well, it is to me, my sister is in a panic over it and shrieks: 'Don't keep him waiting! He hates waiting!' like he's the Queen.
I very rarely accepted lifts from him because of this and would much rather crawl naked over broken glass than put up with the atmosphere.
One of the last times was when he was driving me and my sister and their sons back from the hospital where my mum was busy dying. I seriously wanted to say I'd get the bus and then the train - it wasn't that much of a problem - but my sister insisted and death is a time of togetherness and it would have been awkward to refuse.
We were near where I wanted to be, going along a main road in slowly moving traffic and I expected him to turn left into a side road, let me out and do a U-turn. It's legitimate at this junction and would be a simple thing to do and probably you'd be no more than three cars behind - if that really mattered to you on the occasion of your SIL being imminently bereaved. Any normal person would do it. But he's not fucking normal.
As we were approaching it he said: 'The traffic's coming to a stop. Get ready to get out Limited.' I thought he was joking and said: 'Okay X. Give me the signal and I'll do a commando roll.' His son said: 'Dad's not joking.' The look he gave me in the rear view mirror confirmed that. I jumped out on a muddy verge. What a fucking cunt my BIL is.
The very last time I accepted a lift from him was a day later when my sister called to tell me she'd received a call from the hospital to say we were required urgently and they'd pick me up. I was staying at my mum's at the time. I got washed and dressed in about 10 minutes all except for my shoes and coat and waited. My sister tore up the path and screamed: 'Why aren't you ready? You know he hates waiting!' She then tried to do my shoes up for me like I was five. I told her to leave off and walked down to the car deliberately slowly.
It wasn't 'Hurry up! Mummy's dying.' It was: 'You know he hates waiting!' What a fucking cunt.
Sorry for the hijack OP. The bile against my BIL has been rising for a long time and you triggered it. I've no idea why your woman waits outside the school 