The sign relates to the Southwark neighbourhood area which is adjacent to this bit of Lewisham. This area is right on the borders of the two boroughs. Peckham Rye (address and Park) itself is quite a bit further down Forest Hill Road. One Tree Hill and Brenchley Gardens belong to Southwark. All three green areas can be joined together in a nice green walk.

To largely reiterate what others have said I don't think you have anything at all to worry about. Nowhere is completely safe, but after living near where you mention for close to 25 years I can't say I have ever felt threatened. I walk through Brenchley Gardens often and then loop round into Peckham Rye. Both are very pleasant and popular. If you like a bit of a climb, One Tree Hill is also a great place to sit and ponder whilst looking at the London views and the Horniman is close by too. I think there is a surprising amount of greenery within a 10 minute walk (or less) for somewhere so close to London.

It can be a bit dark on the One tree hill side of the road from HOP station - agreed. If you feel a bit unsure, walk on the built up side. You could always get off the train at forest Hill and walk up from the other end of Canonbie Rd (the Honor Oak rd end) a good work out for the calf muscles!

The estate you refer to has always been fairly quiet. I wouldnt be surprised if quite a few have been privately purchased (great views of the cemetary on one side and park on the other!).

At the end of the day nowhere is totally 'safe'. But I was very pleased to be back in Forest Hill a few weeks ago having taken advantage of the Overground, visiting with interest 'the other side' of London.