Hmm... ok, I have some free time so if you want to PM me details of an admin account for your forum I'll see if I can find any problems.
In my browser (Safari), I had activated a stylesheet which disabled ads on web pages. It was this that was preventing me from seeing the "New Thread" button. Deleting that stylesheet solved the problem. Thanks to Matt, who selflessly spent time trying to get to ther bottom of this.