My advice would be to plant extra of something(s) in your own space and share the bounty with your neighbors to get the conversation started. That way, they'll see that 1) it can be done, 2) you care enough to do it on your own, and 3) you're a nice person. Reciprocity should help take over after that...and if someone's not interested just ask if you can "borrow" a corner of their yard :)
When you say your neighbors have gardens, do you mean vegetables? Flowers? Something else? Or does the term garden in the UK mean something more like yard or lawn does in the US?