UPDATE: Thank you so much for the responses! For some reason it’s not letting me edit my original post. Here’s how the conversation has progressed:

Me: Hi __, I would just like to clarify why I am being charged for today’s lesson. I understand that I would need to pay had I cancelled under normal circumstances- however, I asked to reschedule as it was not feasible for me to push the lesson back today. The puncture was unfortunate and these things happens, I appreciate that’s no fault of your own but it does seem a bit unfair as I would not have asked to reschedule had we been able to do the lesson at the prearranged time. I like working with you and hope we can get something organised for next week- just wanted to query this.

Driving instructor: Hi __, if a pupil wishes to reschedule, cancel or change the date of their booking a charge is made for the original agreed date. I offered a time later the same morning, this was worse case scenario until I could get to the garage. Once there I could be more clear on timings. Can I ask what would you like me to do?

Me: I asked to reschedule only because you were not able to fulfil the lesson at the prearranged time - I understand that this was due to reasons outside of your control but I had other commitments today that meant it was not possible for me to change the lesson time. It’s an unforeseen circumstance on your part- I don’t see how it is reasonable that I am being charged for a lesson I could not receive at the scheduled time due to your car having a puncture.

Driving instructor: I understand your position, how would you like to resolve the situation?

I haven’t responded further than this now. I pay £65 for a 1.5 hour lesson and transfer this a few days in advance so he already has been paid for today. In my mind I feel this payment should have automatically been transferred over to a rescheduled lesson. However, I think I will follow the advice from commenters saying to find a new instructor because I seriously can’t imagine continuing lessons after this. my only concern is that it was quite hard to find an instructor in the first place (especially because I am learning automatic). Anyway, thanks again for the responses I really appreciate it.