When I've had a lesson at instructor's yard she insists on both a hat and a groom because it's her insurance. When she comes to me at a friend's yard, then it's up to me/my friend, our risk.
For many years I drove on the open road without a hat as I loved bowling along with the wind in my hair until I was bolted in carriage (not my pony thank goodness). This mare kicked the cart to pieces and my instructor and I were left with a rein each and nothing much else. After that I never drove without a hard hat. If I had hit the road on my head I am sure I would have been killed. So do take care of your head it is the only one you have!
IMO its exactly the same as being in the saddle, you're stupid not to.
I would expect to be asked to wear a hat if I was having a driving lesson; and if going on the roads hi-viz as well. I know some people who drive who'd also advise you to wear a BP.
I recently did a carriage driving course (it was ace, I loved every minute). We had to wear a hat whilst we were in the carriage, they also recommend gloves, string, leather or pimple palm type will suffice. After the first day (I did a 6 week course) wearing my good Charles Owen velevt show hat, I went to decathlon and bought a hardshell type; better ventilated, and less inclined to attract all the cr*p when you put it down on the floor when putting to etc. Cheaper to replace as well!