Well, this is the response that I got from one of the technical support people:
Paul
The problem seems to be that you have got so many posts. You need to go into the administration and prune some of the sections. I
stopped counting how many posts you had after 30,000. That is way too many for the database to load quickly.
Well, I've set up a blank test forum on the same server and tried posting and editing there and yes, the speed is better.
But as everyone will remember, the forum as it is (this current existing size) was going at a totally acceptable speed 99% of the time on the old server, so I can only conclude that Li'l monster may not be up to the job. There are way larger phpBB forums out there on the web, and I need something that will be able to grow with BaO and not restrict growth. I'm therefore going to have a final try at seeing if there's anything the server folks can do - if not, it looks like I'll be searching for another server again. If we do have to move again i'll try to ensure the downtime is virtually zero (unlike before).