In the years following WWI, the Police determined a need to oversee the demonstrations in Trafalgar Square. They originally wanted a normal police box but objections were raised to this and so the plan was drawn up to convert a granite column at the corner of the Square into an observation post for a single constable who had a phone link to a nearby station from whence he could summon help if required.
The corresponding pillar on the opposite side of the Square shows what it originally looked like.
The box passed out of use with the advent of mobile communications and is now used as a store by local authority cleaners.