I took my car to the dealer who tested the adapter with another iPod, which worked fine (I already knew this though). They then rang Honda who confirmed that the adapter should work with all iPods, assuming the connector was the same - this was news to me (pleasant news, that is).
They suggested that my iPod was faulty and that I should go to Apple and get a replacement, which seemed a bit unlikely. The iPod works fine with my PC and with my Bose docking station etc.
I had already done a 'restore to factory settings' from iTunes and thought that was about the end of it. However, heading home I had the bright idea of performing a hard reset of the thing, by pressing the middle and top of the dial simultaneously (as detailed on the Apple site). Suddenly I got music coming out of my sound system which sounded strangely familiar!
Initially, I couldn't believe that it was the iPod, but skipping through a few tracks proved that it was!
I have now added half a dozen playlists and it's working perfectly.
Just goes to show - the iTunes reset isn't as thorough as I thought it was!
All's well that ends well - I am now very happy with the result - despite all of the complaints on this forum!