We'll have single payer healthcare before we have "driverless cars", because the insurance industry lobby will prevent it. If everyone had a perfectly-run, accident free, computer controlled vehicle that obeys every law and will always avoid an accident, they'd have to lower rates. And you *know* they aren't going to do that.
They may have to lower rates, but that doesn't mean it will lower profits. As long as their costs (payouts and administration) go down faster than they're premiums their profit could actually increase while lowering rates. Insurance companies don't like paying claims, so you can bet as soon as autonomous cars get safer than human drivers, the rates for human driving will go up and will go down for autonomous cars. The faster autonomous cars get safer, the faster the rate differential will create the incentive to move to autonomous cars.
So while some claim that it will be even more than 20 years because current cars will stay on the road, instead it is possible the insurance and liability incentives will cause human driven cars to be so expensive that the transition will happen faster. Of course, those people could be right because it's hard to predict the future.
I also think that it's possible vehicle to vehicle communication will have a bigger impact faster than fully autonomous cars. If cars can relay information about hazards and make new maps, then cars don't have to be fully autonomous to make them radically safer. On the other hand that will take good communications standards and there's a good chance commercial pressures will cause the possible gains to not be realized, similar to the possible gains of electronic medial records not being realized because of the failure to ensure different systems cooperate efficiently.