Would you like to know something wonderful about London and Middlesex? They were used almost interchangeably up until 1965 (and even somewhat now) except for in the very small 'City of London'. In the strictest legal sense, Greater London and therefore St George Hanover Square was 'Middlesex' pre-1965 but it's also important to consider that to many people 'Middlesex' had no meaning. In the eyes of most they were Londoners through and through and 'Middlesex' was their government branding.
In essence, Middlesex represents a defined boundary that was usually sometimes used by the government and London was an identity that people stood behind. So it's your choice: St George Hanover Square was London, Middlesex and both, all at the same time to different people.