Deserter has put together a map of pubs around South East London that don’t require booking. It’s a bit thin, but possibly of use none-the-less. Locally it lists Blythe Hill Tavern and The Dolphin, Sydenham.
We also know that the newly re-opened Perry Hill pub only requires booking for covered tables, and while booking isn’t necessary at these venues I guess it doesn’t ensure you can find a space at busy times:
No booking required for the All Inn One, just #bringablanket 
No bookings for the Spoons of course - the Capital, and slightly further afield the Brockley Barge. I haven’t tried them as yet. Neither of them have particuarly large outdoor areas, so not sure how many people they can squeeze in.
I went to the Dolphin first time ever and it was great met my friend for a catch up and my gosh… Yep… I be going back again not sure when but lovely staff. Another good local added to my list of go twos happy days 