I've been hit by uninsured drivers twice in the last 6 years, both in London, and both caused a HUGE amount of hassle.
Report him. Immediately. Now. No, really. Give the police ALL the details he gave you. And make it quite clear that you will be 'on their case' to find out what happens to it.
I would bet on the fact that the name and address he gave you are wrong. The Police may (may!) be able to trace him from the car details. This is your only way of being 100% sure you have the right man.
He is also VERY unlikely to come up with any cash whatsoever. If I was you, I'd give him a week to give you the whole lot, in cash. Yes, it's a big ask, but if he doesn't then start a court claim against him.
Yes, you may frighten someone who would otherwise pay into not paying. However, the chances of him having the money and the intention to cough up are very small, and you need to chase him as hard as you possibly can.
It may be worth talking to your insurance company as well, to see if they can give legal assistance without affecting your NCB.
Oli.