When Cameron first called the referendum Remain was in the lead. When topics are debated thoroughly in the public sphere and serious campaigns are run, people's opinions can shift pretty dramatically. That's why politicians campaign.
Nobody has ever spent time campaigning to keep railways privatised in the UK because the Conservatives have always chosen to fight elections on other issues, whilst Labour have made nationalisation a priority for years. If people were asked to make a direct decision on this and there was competent campaigning involved, I am very sure nationalisation would lose. The arguments are weak.
87% of Britons polled said that they are very proud of [the NHS]
The same poll showed even more people are "proud" of the fire brigade, although there's nothing special about the British fire service. They are also more "proud" of the post office than Oxford or Cambridge universities. All that poll says is that people tend to answer "proud" (whatever that's interpreted to mean) when asked about institutions which they frequently interact with and are rarely exposed to any criticism of.
Just like with the railways, British people are not exposed to serious debate about the NHS. The Conservatives have, for now at least, given up trying to debate it because they prefer to be a centrist party and because Labour consistently exploit people's emotions by conflating the NHS with healthcare. For instance the left will happily imply that any criticism of the NHS (a bureaucracy) means hatred of nurses and loving of cancer, or other nonsense.