I live in SE London too. Frankly it is not the most wonderful place to cycle for pleasure. Green fields and country pubs are in rather short supply and chicken nuggets rather displaces cake and crumpets.
Most of the clubs, like ours, that were set up in South East London when Surrey and Kent were only a few minutes away have followed the expanding borders to remain on the edge of London. What is really good is its so easy to take your bike on Overground/Southern & Southeastern to cut out the boring bits, get straight to the modern start points and get into country riding.
Hence the Anerley now starts from Purley, Sydenham from Chipstead and Catford from Farnborough. Anerley is small and sedate, Sydenham is more time trialling and Catford love 'ills. Addiscombe (South Croydon) is one of the biggest all sorts. Dulwich Paragon and Norwood Paragon are good for young fit lycra luvvers ;-)
For rides exploring the wonders of Lewisham and Southwark check out the local LCC branches imaginatively named Lewisham Cyclists and Southwark Cyclists.
If you do decide to take a spin out with The Anerley (anerleybc.org) just turn up at the start point near Purley Oaks station, don't panic - nobody gets there until 1 second to go - and you will be made very welcome as a guest. The Sunday ride breaks up after the tea stop for an easy return (~ 30 miles) and the drunkards who go on to do ~50 miles. Some old codgers take out smaller rides on Tuesdays & Thursdays.
