I think there would be a feedback loop in place where self-driving cars demand better infrastructure to support them, and as this infrastructure improves, self-driving cars become more convenient to use, leading more people to want to use them, leading to more such infrastructure. Since this infrastructure gets funded, it's not much of a stretch to think that most transport money would go there.
Some of the technology might make public transport more efficient (self driving buses seem like a good idea), but cars would probably dominate.
Some of the technology might make public transport more efficient (self driving buses seem like a good idea), but cars would probably dominate.