If had to try and predict what might happen, then I'd say that it will be city areas where autonomous vehicles will initially have far and away the greatest impact. I think it's entirely possible that we may have virtually no private cars, and no driver-operated buses and taxis in some of our cities within the next ten to fifteen years, maybe less if legislation continues to make driving in cities more expensive, year on year. There will be a lot of resistance, from taxi and bus drivers, but it seems inevitable that this will happen. Having intelligently controlled autonomous vehicles, that can provide safe, quick and reliable transport for individuals or groups of people would become popular in just the same way that Uber has. Removing drivers from vehicles in cities removes a great deal of the problems, like thoughtless parking, the congestion that causes, the need for large car parks full of vehicles that aren't used for >90% of the time, etc, as well as solving the vehicle pollution problem, if the replacement system is electric.
When it comes to transport outwith cities and large towns, things get a great deal harder to predict. I suspect that people like us, who live ~10 miles from the nearest reasonably sized town, might be the last to move away from car ownership. It's quite possible that the personal car market shrinks to the point where owning a car gets significantly more expensive than the alternatives, especially as those alternatives reduce in relative cost, due to their popularity in urban areas.
There are plenty of times right now when, if there was a convenient and affordable alternative to using a personally owned car I would use it. For example, our local pub is about 4 - 5 miles away, so we drive there, which then means one of us not drinking. Going to the theatre (when this plague is under control) is a >20 mile round trip, and in winter, or bad weather, I'd much rather not drive if there was a convenient alternative. Things are already changing, over the past few months we've completely stopped going to shops, and do almost all our shopping online, with all of it, except the local farm shop order, being delivered to the door.
When people talk about disruptive technology, like autonomous transport systems, I think many may not realise just how disruptive it may be. It's entirely possible, perhaps even very likely, that personal car ownership will just cease to exist. The primary things that make car ownership attractive now are the convenience (perhaps the most significant factor) and the effectiveness. The cost of car ownership is already high, higher than the cost of most other forms of transport, in terms of total cost per mile (not just fuel cost). Take a Model 3 as an example. Over 5 years it will probably cost ~£20k to £30k in depreciation, ~£2k to £3k in insurance, and for 10,000 miles/year around £2k in electricity, so around £3/mile. If something else can be every bit as convenient and effective, at significantly less than about £3/mile, then pretty soon most people would just stop owning cars.