Dave
Plummetting is not quite correct, but the 'glide' angle of an autorotating helicopter is considerably steeper than that of an aeroplane.
Conversely, the space required for a helicopter to land after an autorotative descent is much smaller than that required by an aeroplane. The helicopter pilot can flare off almost all of the forward speed before touching down.
The 'specified area' of central London has designated routes for single-engined helicopters, which keeps them over areas judged to be suitable for an autorotative landing, without overly endangering those on the ground. Practically speaking this means that helicopters flying over the very centre of London have to fly along the Thames, and in the event of an engine failure are obliged to ditch in the river.