If the guy pursues this, then you are going to be fined and possibly have your licence endorsed for no insurance, and no MOT.
You can only drive without an MOT if you are travelling to a pre-booked appointment at an MOT station, and as you confirm, you were not doing so.
If you can contact the other driver, and confirm that you will pay the costs for his repairs, and you don't wish to involve insurance companies, you may be able to do that.
You must make every effort to track down your insurance details - if you can't produce proof to the police, they will prosecute you, the burden of proof is on you as a driver.
Sorry to be the bearer of bad tidings.