What a great idea ! Alas I'm in london the weekend after 1st June, but for July why not combine it with the weekend of the Smithfield Nocturne, which is the 5th I think.
Here's a map of Hyde Park showing the cycling routes :-
http://www.royalparks.org.uk/docs/pa.../hyde_park.pdf
and here is one of Kensington Gardens, which is the other side of the road :-
http://www.royalparks.org.uk/docs/pa...on_gardens.pdf