Just saw this article, which includes a visit to The Horniman Gardens, and it made me wonder if you have other suggestions to add to these, especially if they’re local
https://www.newsshopper.co.uk/news/26036620.five-best-south-east-london-places-spring-walks/
A really nice local walk is the Waterlink Way, which runs (mostly) along the Pool and Ravensbourne rivers from Lower Sydenham to the Thames at Deptford Creek. It’s 8 miles, and I usually walk to meet it at either Bell Green or Ladywell Fields. Some decent pubs at the other end too 
Here’s a pdf – the route is well marked along the way:
Another place, just a little further out, is High Elms Country Park. A lovely wooded park to ramble around. High Elms Country Park Local Nature Reserve
good point. I only really use this between Bell Green and Catford, but I should explore the longer route.
that sounds really interesting. I’ve not heard of this before, but it seems only about 30 minutes away (by car) - and the very excellent Coolings Garden Centre is around the corner from there too (I miss having the Shannon’s Garden Centre here in Forest Hill)
I do quite a few ‘station to station’ walks in Surrey and Kent. Yesterday I did this lovely 4-hr walk from Caterham station to Woldingham station via the North Downs Way (both stations less than 50 mins from Forest Hill by train and nearly all inside the 60+ Oyster card zone too). You can shorten it to 2 1/4 hrs by taking the green dotted path up past Woldingham school. Lovely views and loads of wild garlic at this time of year.
For local(ish) woods we’re big fans of Oxleas Wood, Petts Wood and Keston Common, all about 30 mins in the car.
Enysford (30 mins on the train from Catford) is great too, quick to get into the countryside and you can stop at Lullingstone Roman Villa and have a paddle in the ford. Reccomend the Malt Shovel for a pint of Harveys Best.
This is a nice route
Eynsford and Lullingstone circular - Eynsford and Lullingstone circular – Google My Maps
