Their website says:
"The company welcomes everyone who would like to take part in all areas of amateur theatre. It's a friendly and relaxed group; we don't believe in holding nerve-wracking auditions in order for you to become a member."
I have a few friends who go to Network and really like it.
Also - but admittedly a bit out of SE1 - is the South London Theatre in Norwood, which I have now been involved with for about 3 years - it's a friendly place, and with up to 22 plays a year, there's always lots to get involved with - plus it's got a bar as well (which is always a good thing). Have a look at http://www.southlondontheatre.co.uk/
The Inner City Players are a group that has been going for over 30 years. They rehearse and perform at the Lambeth Mission. Each year they put on a smaller production just before Christmas, which include short pieces, reading, songs, sketches etc around a theme. Their main production is a musical which is performed in the spring. The last three have been Me and My Girl, Fiddler on the Roof and The sound of Music. They have been very good procductions (no I haven't been in them but my kids have and I have helped out behind the scenes) It is a very friendly group and is open to all ages and abilities and new people are always welcome. Auditons for the main parts take place in Jan. I have the number of the secretary but as it its a private number I do not want to post it here but if any one is interested let me know and I can pass on or give you more details.