Never a coat on top. Ever.
In the very cold weather he will wear a long sleeve thermal top under a long sleeve shirt with his blazer. Then he wears a hat and scarf and gloves. But never a coat. Sometimes I have managed to convince him to take a compact umbrella if it's pouring down.
For a while he wore a hoody under his blazer for the journey to school, but the school sent out newsletters telling us it was against the rules.