As you say, this is particularly relevant for places that appear to be public but have no signage to indicate it's actually private land (e.g. London Eye, Canary Wharf, Trafalgar Square).

Unfortunately there is no simple answer. Sometimes there are clues, eg ByeLaws displayed in markets, parks, railway stations etc, but quite often there is absolutely nothing. Almost all of the UK is in private ownership, and some of that private land permits and encourages public access. Access is however a matter of discretionary permission unlike public Rights of Way such as footpaths and roads, where of course you will be free to photograph until a policeman decides you are causing an obstruction or might be a terrorist. Access alows conditions to be placed on photography by the landowner where public RoW do not (except for statutory restrictions such as the Official Secrets Act). So if you want to be sure where you can take photos without some jobsworth telling you that you can't, the question becomes 'how do I find out whether I am on a public RoW?'. 

Mostly it's obvious, but when it isn't the complicated and inconvenient answer is that local councils have a statutory duty to maintain definitive maps which show public RoW in their area, and you can ask their Rights of Way Officer for guidance. However, many councils are slow to update  maps, and the information is not always totally reliable. I've found this particularly to be the case where the location of the 'building line' is concerned - the point where a public footpath shades into private property. It's not unusual for one side of an adopted road to be public and the other private. Often nobody really knows, including owners, security guards and traffic wardens, and the answer is buried in the deeds in a bank vault somewhere. There's also the complication that some RoW are evolved by usage over time, eg many rural footpaths have uncertain status. Nobody knows until a court decides.

Having said that, photography on private land to which the public have access is not prohibited unless there is a clear and obvious notice that prohibits it. It only becomes a legal problem if you disregard the landowner's instructions or terms of admission that restrict photography, or flout normal expectations of privacy. Landowners have no rights to demand deletion of photos taken without permission, and no claims against your copyright unless they derive income from their own images of their property. They only have a right to demand you stop taking photos and/or leave their property, and to use reasonable force to eject you if you do not comply promptly.