I did my driver's license test on August 3rd 2021 at New Ottery Traffic Department (Cape Town) and I passed (yipee)
That day you would have had to make a payment to them, correct? Don't you still have the receipt? Or were all systems offline?
Regardless, as you've passed the test, you're a legalised driver from that date onwards, irrespective of the presence of a card licence.
Start the claim procedures, and supply the card licence at a later stage when required.
However, if you do not have a police report case number for that particular collision, you may be out of luck.
Also, the other party could claim that YOU were in the wrong. They could even counter-sue, but they'll have no legal grounds to stand on, unless they obtained a licence in the past. (Again, any card licence is only an indication of the existence of a driver's licence obtained/granted, so to speak.)