Hi Roundouthigh,
You stand a good chance of flying your detail on Saturday afternoon, provided you don't mind sharing the airspace with pleasure flying helicopters, Helimed, Police helicopters, Classic Wings Rapide, PA34 from Stapleford, and the dreaded "Dome watchers." Also, subject to any last minute special events, photographic surveys etc., etc. Depending on runway configuration at Heathrow you should be able to fly up to a North/South line through Battersea Heliport at 1500ft, and in the City Zone between 1500ft and 2400ft. Suggest you don't fly over any palaces or prisons.