This article references the likes of restaurants and football grounds as private spaces as far as filming is concerned. Unless you can find something to contradict that, the railway is private property in much the same way.
Either way, there is no reasonable expectation of privacy on most parts of the railway that passengers are allowed to access (i.e. stations and trains) and so filming as well as the posting of such films is lawful.
You still have a reasonable expectation of privacy, but the definition of "reasonable" will reflect the situation. If someone was filming your laptop screen over your shoulder you might have a case if you sued them.
That article (with its amusing footer full of search-engine fodder) is somewhat inconsistent with a lot of other opinions I've seen given on the topic, which hold that places to which the public are routinely granted access - whether in public or private ownership - are places in which a person cannot have a reasonable expectation of privacy
from observation.
This has obvious caveats (toilets, changing rooms, etc etc), and some grey areas (like the exact extent of knowing exposure to public gaze that defeats an expectation of privacy on private property), but I've certainly never seen anybody other than the author of that piece advance a realistic argument that merely stepping onto the "private" side of the property line creates a reasonable expectation of privacy. It would be rather difficult to convincingly argue, for example, that a person standing one metre inside the bollards in Victoria station's forecourt has a reasonable expectation of privacy when a person standing one metre outside those same bollards has no such thing.
The example of filming a laptop screen over its owner's shoulder is a good one though, because it potentially transforms simple observation into surreptitious acquisition of personal information without consent, but even still the filmed person would have to pursue a civil case to recover any damages and most of the legal protections apply more to the dissemination of information acquired this way rather the acquisition in the first place.
Even the fact that a person is trespassing to film doesn't necessarily open up protections for people who don't want to be filmed, as only the party on whose property the trespass occurred can pursue damages for the offence.
An illustration of the position of English law on the subject in somewhat-recent times in the case of
Kaye v Robertson [1991] FSR 62, [1990] EWCA Civ 21, where journalists from the Sunday Sport posed as doctors to photograph actor Gordon Kaye while he recuperated in hospital following surgery. A friend of Kaye sought an injunction against publication and an order seizing the photographs and associated materials from the paper, which was granted by the High Court but substantially overturned by the Court of Appeal for the simple reason that "in English law there is no right to privacy, and accordingly there is no right of action for breach of a person's privacy". Lord Justice Bingham went as far as to say that Kaye's privacy had been the subject of a "monstrous invasion", but that "it alone, however gross, does not entitle him to relief in English law."
That said, the passage of the Human Rights Act 1998 and by extension the application of the European Convention on Human Rights to the UK has changed this somewhat. A pertinent example is the European Court of Human Rights' judgement in Peck v. the United Kingdom in 2003 (
overview here,
full judgement here), which found that Brentwood Borough Council had violated Mr Peck's right to respect for private life by releasing to the media without his consent CCTV recordings of him being detained by police whilst experiencing a mental health crisis, and in particular doing so without obscuring his identity. As a result, Mr Peck's actions "were seen to an extent which far exceeded any exposure to a passer-by or to security observation and to a degree surpassing that which the applicant could possibly have foreseen."
Nevertheless, there is still no blanket right to individual privacy outside the home, and the remedy for potential breaches of the rights to privacy that individuals do have must still in the vast majority of cases be sought via the civil courts. It remains how to be seen how Parliament and the courts will manage this area in the future.