Last week I flew a single through London using Banstead, H7, Barnes, H4, Isle of Dogs and depart South along the edge of R158. Saturday morning, to avoid inconvenience to those attending the Globe Theatre.
This is all Radar Control.
Initial clearance to London Bridge, and a short hold (dumbbell-pattern over the river centre) to deconflict with 'big tin' coming in to City Airport.
The map you link to does not have the maximum altitudes marked. Along H7 you are quickly restricted to below 1,000 ft amsl. Along H4, after Battersea Heliport, that ceiling starts to rise and you have to be above 1400ft amsl to avoid R157 (Central London) and R159 (Canary Warf), though you can be sometimes given 1400ft to maximise deconflict with City Airport.
It is a great trip with fantastic views.
Photos from a previous trip here
London heli route December 2010 before the snow...