It depends upon the area and the kids involved.
You look at the kids who live in the arse end of nowhere (like my DP did) and they will have warm outerwear. Because stuff 'looking cool' when you could end up with frostbite waiting for the bus. Or, in recent years, the Polish/Eastern European kids. Hypothermia is not a fashion statement.
But the ones who got driven to and from school didn't need a coat. So they wouldn't wear one.
In other areas, if you're too poor to afford a coat, the obvious answer is to rip the piss out of the kids who have one. Embarrass them into freezing their arses off as well. Or they've never actually owned a coat, so they feel strange and restrictive.
Or you get the shitty looking Roadman coats. Cost a fortune, but all they do is make the wearer look like a fly in a binbag. incredibly desirable though - to the point of wearing the fucking things in the middle of last summer's heatwave.
I think that, rather than insisting upon no coats, no hoodies, a nice coat must be worn, etc, each school should subsidise the purchase of a very warm hoodie so that they are cheaper than a coat. Make sure it has pockets that are fleece lined, make sure that it will keep in as much heat as possible. Make it in a neutral colour, but most of all, MAKE IT AFFORDABLE - even give it away. And stop fucking banning them from wearing t-shirts underneath their shirts.
No child should have to be cold because the only way of accessing warm clothes is to break the rules - and making being warm unattainable for a significant proportion of the population is a guaranteed way to create inequality.