No 1, London (Apsley House, Hyde Park Corner - the Duke of Wellington's residence). All distances are to where the toll bar at Hyde Park Corner once stood.
The 'knowledge' for London cabbies, which was introduced in 1851 after cmoplaints that cabbies didn't know where they were going, states that cabbies must have a detailed knowledge of routes within a six-mile radius of charing cross - giving more credence to Lodekka's answer that the centre of London is generally considered to be charing cross.
For mapping purposes the centre of London is and has been for centuries the theoretical position of the toll bar (long since gone) at what is now Hyde Park Corner. as someone has given above. The cross outside Charing Cross station is a comparitively modern position, its original position was slap bang in the centre of what is now a very busy intersection to one side of Trafalgar Square.