The American expression "Downtown" puzzled me for a long time. I think it's what we call the "city centre". Of course, that reminds me of another strange usage in England. "The City" (short for the "City of London" aka the "Square Mile") is physically the oldest part of London, site of the original mediaeval walled city. Most of its buildings were destroyed in the Great Fire of 1666. It was rebuilt and then a good part of it was demolished by Hitler's bombs. There are still, however, some mediaeval survivals.
But "the City" is like "Wall Street" and can refer to the financial workings of the country as well as to a real location.
The "
City of London" is quite distinct from the rest of London. London as a whole has its own Mayor and its police force, the Metropolitan Police Force ("the Met") with its headquarters at New Scotland Yard. The City has its own independent City of London Police Force. It has aldermen, a hangover from the middle ages, and liverymen, representing its livery companies or guilds. It also has a Lord Mayor, whose official residence is the
Mansion House.
Every year a new Lord Mayor takes office and there is a splendid procession called the
Lord Mayor's Show. It is a proverbially lavish display. In the days when everything was horse-drawn there was a saying: "After the Lord Mayor's Show comes the dung-cart", signifying something along the lines of "pride comes before a fall".