Thanks for the reply.
I couldn't figure it out, so I ended up deleting all the files off the server and trying JoomlaPack (
http://www.joomlapack.net/) to backup my site and then automatically set it up again on the live server. It worked perfectly and ironed out a few other issues I was having - I would highly recommend it.
However, it didn't solve the base url problem. So, I turned off SEO friendly links and mod_rewrite and this has worked, although now I've got ugly long urls like '/index.php?option=com_user&view=register'.
It must be something to do with a setting on the server (it's a shared platform so I can't change anything), or the .htaccess file?