The best thing to do is check the definitive map of public rights of way. These maps are available for public inspection at the offices of local surveying authority although you may be required to book an appointment. Some working copies are available online. Ordnance Survey Explorer and Landranger maps derive their information from the definitive map but you should talk to the rights of way officer at the relevant authority if there’s a discrepancy.
Some rights of way are not yet shown on definitive maps. These paths can still be used, but an application needs to be made to the surveying authority so that they can be added to the definitive map.
Inner London borough councils are not required to produce definitive maps, though this does not mean there are no rights of way in inner London. To check whether a path is a right of way which is in inner London we would recommend that you contact the council of whichever London borough it is to see whether they believe it to be a path maintainable at the public expense. They hold (under section 36(7) of the Highways Act 1980) a document called the List of Streets, which is a list of all roads and paths maintainable at the public expense. If the way is so shown, that means it is a public way of at least pedestrian status and proves that the council has a duty to assert and protect the rights of the public to its use and enjoyment, under section 130 of the Highways Act 1980.