Tragedy of the commons
Of course, any massive expansion in Uber cars at the expense of public transport will cause more congestion on the roads - a shared public resource - and then none of Uber's services, or those of the black cabs, will work well. So there's another limit to the size of the market, unless we want to build twelve-lane expressways through London. And that investment in the public resource is a lot more expensive (leaving aside its unacceptability to residents) than paying a little extra to get in a car instead of on a bus because you feel richer.
You already see this in city centres where deregulated bus services have resulted in a logjam of half-empty buses in different colours, trying to compete for the same lucrative routes. Obviously you can't get a bus at all on the less lucrative routes any more because even the councils have given up subsidising them. Mind you, you can't get a black cab to take you sarf of the river after chucking-out time, either.