To replicate that, though, Uber would also have to replicate some of the state apparatus around it, which they may or may not be able to do (or even want to do, ideologically). The London exam is effective in part because it's backed by law: it's illegal to drive a taxi if you haven't passed a licensing exam, and it's also illegal for someone who is properly licensed to lend out their credentials to a friend who'll pick up some people on their behalf, while the licensed driver takes the day off. And as you make the exam more rigorous, those counter-pressures increase, so if Uber had an actually rigorous knowledge exam, they would also have huge levels of cheating if there were no attempt at serious enforcement.
Uber could try to institute both a rigorous knowledge exam, and a high level of enforcement of the rule that only people who actually passed it may drive cars under Uber's name. But they would need some credible way of doing that, like roving undercover inspectors.
You could have the app send you a picture of the person who should be picking you up. The user could then report if someone else comes to pick them up. The only problem I'd see is that it might make riders uncomfortable if it is someone different, they might not want to cause a scene or any sort of confrontation, and they want to get to their destination, but they also wouldn't want to ride with someone unlicensed.
So I'm not sure exactly how you'd get around that problem. Maybe you give people who report that they have a different driver say $20 credit, and the ride is free or something. Of course then you'll have people trying to cheat the system.
