If the private car was banned today, the public transport network wouldn't be able to cope with the additional demand.
That is true in the short term.
In the medium term you would see a surge in bus operators which would rectify almost all of the problems caused, and possibly higher investment in buses from local councils could lead to most of the rest of them being resolved.
In the long term you would see a major restructuring of towns up and down the country to eradicate car-centric elements, to the benefit of just about everyone.
Of course, there would be a very small number who would experience serious adverse impacts from a ban on cars even in the long term, and I do not see it ever happening, or even advocate it. However, that doesn't mean that we can't do the restructuring anyway, and there have been some positives, such as the growth of pedestrianised streets in city centres, park and ride schemes to get cars out of city centres, congestion charge schemes, and the slowing of the expansion in out-of-town superstores and gradual rise in bus provision to edge-of-town ones. This as well as the slow growth in light rail networks and the rather more rapid growth in cycle paths are all positive steps.
But there is a long way to go yet, and projects are still being approved that are steps backwards rather than forwards. Plus most of the more useful improvements are in urban areas - rural areas are still seeing drops in bus services, and investment in roads with little consideration for pedestrians and cyclists, along with a gradual decline in local shops. This matched only against a growing cycle network that is more for leisure than for actually wanting to get from A to B, and a handful of traffic calming measures that are fought against by the over-powerful road lobby at every turn. Plus gradual improvements in rail services, but this is fairly insignificant as most of rural britain is completely unserved by rail and will remain that way.