So, a hard brexit will result in my no longer being able to drive in France. Here's how it goes.
At present, as someone from an EU country, I don't have to exchange my driving licence for a French one. I can drive with my UK licence. In the case of a negotiated exit from the EU this is expected to continue.
However, a hard brexit will mean that I will have one year before I need to exchange my licence for a French one. We expect to leave France to live in the UK in 2025, so that means lots of administration for the sake of driving occasionally in France for five years. Then exchanging it again on our return to the UK.
I won't bother.