While I cannot speak for the Vatican City, I can, as a resident here in UK, speak about the City of London.
Sir, you are wrong in your assertion that the city of London is somehow 'outside' the country in which it is located.
Please read - 'The
City of London is a
district of
Greater London. The city's boundaries have remained almost unchanged since the
Middle Ages. Although it is now only a small part of the
metropolis,
it is a notable part of central London. The City holds
city status in its own right, and is also a
ceremonial county. Often referred to as
the City, or the
Square Mile, it is just over one square mile, 1.12 sq mi (2.90 km2) in area.
It is here in the City of London where most of the United Kingdom's financial
trade is done. It is a very small area, only a square mile, and has a very small resident population (8,000). However, many people come to work here and during the day it can be very busy, with some 300,000 people in it.
"London" now refers to Greater London, which is made up of 32
boroughs (including the
City of Westminster and the City of London). The city is in central
London and is the oldest part of the city, dating back to
Roman times. The City of London has its own mayor, the
Lord Mayor of London, an office separate from (and much older than) the
Mayor of London. There are other ancient features, dating back to medieval times. The city has its own police force, the
City of London Police.'