I think you have to recognise that our roads were mostly laid out in the days when there were very few cars and lots of people walking (and pushing big prams). ... So, in my view, there is a strong argument for giving a bit more space to cars and a bit less to pedestrians than was appropriate in 1900, so long as - and I mean this sincerely, the remaining pavement is wide enough for double buggies, and for pedestrians to walk safely. - andrewr
I would have thought there are far more people using the pavements today than when these roads were first laid out, given the far greater density of people in the area now. Only a minority are car owners.
Plus it would not have crossed the designer's minds that the whole of the road space would be out of bounds to pedestrians. 2 abreast pavement width is fine if those in a hurry, those with dogs, groups of children etc could walk for long stretches in the road.
The pram theory does not make sense either as I doubt they were much wider than a double buggy - they still had to be manoeuvred through doorways.
There is a good case for widening the pavements and perhaps installing a cycle lane, which might be possible if the road were made one-way.
It is offensive to suggest narrow pavement space should be reduced to make more space for rat runners, but this is happening by default especially on the railway side pavement. 
They should be towed away.