Yes, Epping Forest - West Essex might be a good bet - there used to be campsites at Loughton and that part of the world (on tube from London).
Not camping but you might want to consider joining the YHA -
UK Youth Hostel Association (YHA), Budget Accommodation & Adventure Activities - YHA - as well as hostels (there's one in the Lea Valley near Waltham Cross) they do camping barns ...
Personally, if I were in London, I'd hop on the train and go camping in France!