The following are initiatives offering free sport for children, they may not be specifically for London but all of them offer courses/sessions in London.
Asda is currently running a community sports initiative that gives free access to children at various sports facilities. You can search for what’s available in your area here
Not to be outdone Tesco is also offering free football coaching in February 2011 half term so signup for the email now here to be notified of where and when the sessions will take place.
Lastly, there is a new charity to encourage young children to play tennis. You can play for free and also get free coaching, check out the webiste to find a location near you.
As usual let us know how you get on, or if you know of anything similar.