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Here's a nice map app, showing all the soil types in Britain.

SE23 is a bit of a disappointment though.
The bedrock is all london clay - I thought our hill might have some sinister feature forcing its way up from the depths of hell, but apparently not.

The site does not specify what the superficial layers are - presumably more of the same - London clay.

The exceptions are the Perry Vale areas where the superficial layer is labelled 'HEAD' -CLAY, SILT, SAND AND GRAVEL, which might explain a mysterious patch of gravel I found in my garden.
I think it true that this head layer is mainly found in the lower areas of se23 except for a patch under the high lands of forest hill boys school.

The other exception is the bedrock of at the top of sydenham hill/elliot bank - this is a more recent layer of claygate - SAND, SILT AND CLAY. I do not think it right to say this layer is the reason for the hill, but it might be more correct to say the hill is the reason this layer did not get washed away so easily. I'm guessing!

It would be interesting to compare this to a local contour map, but I dont know of any online detailed contour maps.