Most of coast of Essex is mud flats, with some crumbly mud cliffs at Walton on the Naze - definitely not suitable for climbing.
BMC produces a guide to climbing walls in the UK, both indoors and outdoors - see http://www.thebmc.co.uk/Feature.aspx?id=2234 Download whole country or region(s). Lists climbing walls in Essex at University of Essex & GoBananas Play, Colchester; Harlow Outdoor Education Centre; Clements Hall Leisure Centre, Hockley; Stubbers Outdoor Centre, Upminster; Fairplay Outdoor Education Centre, Witham.
Bear in mind that all the climbing in Kent is on friable sandstone, which means that you can only top rope the climbs (you're not allowed to place gear on lead anywhere). You will need a static rope to set up the top rope, plus a long sling (the top rope shouldn't be in contact with the edge of the crag). There's a good new guide (Southern Sandstone or something like that). It's a bit different from traditional lead climbing and the majority of the climbs are technically quite hard (5a onwards) and short - i.e. 10-15m. A useful tip would be to bring a doormat for brushing your feet as your start any climb, otherwise because of the sand you will have b*gger all grip.
Personally I would go to the climbing walls in London, either Mile End or the Castle in Stoke Newington - both of which are excellent, and then try to go to the Peak District for the odd weekend. Somewhere like the Roaches in Staffordshire is only a couple of hours plus from London and the climbing is outstanding (try the Sloth...).