See The Unofficial UK Climbing Home Page for better directions & descriptions.
There are @ least 10 sandstone formations in area around Tunbridge Wells which is less than an hour S-East of London. Probably the best climbing is @ Bowles Outdoor Center, where one can climb without hassle all day for 1 pound 30. No leading, no rapelling due to sandstone. Brilliant top rope anchors and descent path. Bathrooms,coke & coffee machines on site.
Other sites: Harrison's Rocks is not worth the long hike in (how can you say that??? Al's edit ) but has climber-sponsored camping area w/ toilets & running water (even hot) for 2 pounds/night.
High Rocks is impressive & pretty but is mostly wet, mossy or high grade climbing. Also, it is owned by a pub & can be a problem to get in early mornings (unless you climb the fence out by the parking lot).
A less popular climbing site in Eridge can be used for covert camping.
Bulls Hollow, Basset's Farm, Stone Farm, Under Rockes are all worth a visit - see the Southern Sandstone site.