As you are probably aware, Mayfair's name comes from an annual fair that took place in the area that is now Shepherd Market. This rowdy event lasted a fortnight and fell out of fashion as the area became gentrified.
I was interested to read on the alternative media IndyMedia.org.uk that an attempt was made to mark May Day - International Worker's Day - with a revived "May Fair" this year with a gathering of about a hundred people, some of them claiming to be "recognised enemies of the state" in Shepherd Market. There was music, dancing, a play and even someone in a Boris Johnson masked placed in the stocks. From the accounts I have read on the site, the event was well policed and it passed off as a peaceful event.