It's perfectly appropriate to text him again. Nothing rude at all.
No need to worry about asking again. Do as if it was your 1st time asking, no need to mention your previous message either.
Hi [ John ], I wonder if I can have a lesson on Monday 31st ? What time would be fine with you ? Can you please confirm ? (You can also ask for the lesson according to your time : I'm available at 10 AM, 1 PM and 4PM, does any of these hours fit your schedule ?)
If the teacher is often very busy (as I am), and doesn't answer the phone when teaching (as I don't), he may have forgotten to answer (a 5 days delay means most probably he forgot about it though). Better for both of you if you send the message/reminder at least a couple of days before schedule.
You can also send an email (many have website and contact-form for that purpose, sometimes an online schedule that you can access and book from). This works for me (in case I forgot to answer the phone), because I read them and answer :
- very early (around 8 AM) before going into the car.
- at lunch time (if any ^^).
- late in the evening, after closing the school.
Side note: 20 years working in a driving school, if this matters. Not in UK, but the answer is most probably OK anywhere else I guess...
Funny fact: I once received a message 8 days after it was sent, student showed me on her phone :)
Drive safely, never text or answer the phone while driving, and all the best for the final test :)