Short history: approximately free market in rail until 1913, built mainly with private capital. Government control/direction during the war. Government decides the railways aren't making enough profit in 1923 and reorganises them into bigger regional monopolies. These aren't very successful (in a very difficult macro environment) so it nationalises them—along with everything else—in the late 1940s.

By the 1960s the government runs railways into the ground to the point it essentially needs to destroy or mothball half the network. Government re-privatises the railways in 1995—at this point passenger journeys have reached half the level they were at in 1913. Within 15 years they've made back the ground lost in the previous eighty.

But maybe it's not privatisation that led to this growth. Let's consider some alternative hypotheses:

Was it a big rise in the cost of driving?