You could phone the Thrive garden in Battersea Park. A gardening for the disabled project which managed to have the key Pimms ingredients (I now forget which flowers you need for the trad recipe) when I needed it.
Another place with quite an impressive cottage garden is the Garden Museum on Lambeth Road. If they have raspberries you can find the blackberries on the, as yet unbuilt, Founders Place site to the north of Archbishops Park.
Roots and Shoots would be another place to try. I will also phone a friend who helps maintain a school garden and who has an allotment.