Hi MSJ,
I got it working in the end, your suggestions gave me a couple of things to try I hadn't thought about - brain is getting a bit tired after my new adventures in Linux today

Firstly, the main thing that was missing was that mod_rewrite wasn't enabled in Apache. Because most of my URL rewriting was, apparently, working I didn't think that it might be disabled. However, as soon as I enabled mod_rewrite the bits of the web site that weren't working sprung into life.
I had another issue with some of the administrative parts of the web site that weren't working correctly. This was down to file permissions. In my case, applying the recommendations from here -
https://www.drupal.org/node/244924 - did the trick. Effectively, as I'd moved across from shared hosting, everything there was permissioned with the owner and group being the same, i.e. the account that the shared hosting provider had assigned to me. Now that I'm on a private server, I had to change the group to www-data. With that done, the permissioning issues went away.
So, all looking good now, I think! Have put a couple of low value orders through the web site and everything seemed to go through correctly, so looks like it's all running well.
Thanks for your help,
Mark.