Try GoOutdoors
Huge selection they link with DofE
Try some coats on as sizes can be inaccurate if you want to go over blazers etc.
It's a gamble because kids lose stuff at school, and getting a very good waterproof won't be the cheapest coat in the shop.
You pay for waterproof items to be lightweight and pack small, so it's a balance between cost, size, how waterproof, what temperature it will go down to
How much do you want to spend?
How trustworthy is the teen?
But GoOutdoors is a good starting point,lots of choice, range of budgets.
So the Peter Storm range there is a £35 jacket with 3000mm rainproof, and a £75 5000mm rainproof which is the amount of hammering the fabric can take before it leaks basically.
The best rainjacket I had was gortex, it was £100 at the time, no lining just waterproof and oh my lord was it waterproof lasted 15 years with much use, similar, double layered is £160 (I obviously want one now).
I'd go and talk to someone at an outdoor retailer. They will be able to help you.