Seems no one has anything to say on this, so here's my take on it.
Assuming World War 2 hadn't happened and the railways hadn't fallen into such a dilapidated state, and Labour didn't win the election, they Big Four would probably have survived, at least for a while. By the 1950's travel was changing, with an increase in road freight competition and the rise of the private car. Could the companies have competed any better than BR? Railways had to compete on a national scale as far as freight went, to keep the long haul stuff and let local distributors deal with the town areas. The trouble with the smaller depots like Darnall and Attercliffe was they were incredibly labour intensive, and costs were rising, so pushing the traffic onto the roads where fewer people were involved , hence lower costs. With the onset of containerisation new facilities would have to be built, but would each company want to go to that expense? Tinsley was already pretty redundant by the time it opened as the freight traffic had changed in just a few years. The electrification scheme started by the LNER would probably have been extended to London, there was originally a large order for the EM2's which was cut back to just seven, so it was obviously on the cards.
How long the companies would have lasted is another matter, falling passenger numbers, falling freight loads, all due to road traffic, would probably have led to amalgamations and takeovers, and assuming the Government didn't get involved, we would probably end up with a couple of large private companies, like First and Stagecoach. Non profitable routes would have to be cut (just like local bus services are these days), so some rationalisation would be inevitable.
It's difficult to say if Sheffield Victoria would still be with us today, there are so many variables, remember its staple traffic was coal, and that's gone, passenger numbers from the intermediate stations wouldn't be high enough to keep it viable so we are left with just the through traffic from Sheffield to Manchester. The alternative via the Hope Valley has the stone and cement traffic to keep it viable, so the Woodhead would probably been closed, but probably not as early as the 1980's.
It's all conjecture, and others may have a different take on it.