It is true that it can be a bit of a race track at times. Many times at the Zebra crossing cars whizz past with no intention, or indeed likelyhood of stopping.

Maybe some shared space solutions would work nicely here too, cf:
Traffic lights Dartmouth Road/London Road

I am thinking junction of Honor Oak Park/Honor Oak Road/Forest Hill Road, Honor Oak Park section Devonshire Road to Grierson, perhaps even Stondon Park junction too. Actually, why not whole shopping parade?

This is relevant too:
Complaining about traffic on Forest Hill Road

I suppose the wrinkle here, at least going up the hill, is that there are 2 councils involved. One for the neighbourhood plan?