I asked the same question recently on here, and was advised to go the The Manna Centre, 6 Melior St, London SE1 3QP, which I did - they were very grateful of my clothing donation. Its just behind London Bridge station
There are clothes charity shop collection points on Shad times for Scope. I also go to Mind and Sue Ryder on Southwark park road (so not too far walk from you) and mix a trip in with going to the blue market, Vinh Pah and St James Turkish Supermarket. Sue Ryder have been great and sent me a thank you note afterwards updating me on how much my donation had raised for them and what this meant! They take all sorts, Mind didnt take books as "no-one reads round here do they; they are all into texts n that?" Sue Ryder took the books.