I think I've found the root of the 'slowness' problem. I've been monitoring the performance and it turns out that the server can no longer handle the volume of traffic coming through the site! It's fine until the server needs to do something else, like perform a backup etc, at which point it all goes haywire. It's several years old now, and is the last of it's kind in the data centre
Unfortunately, at just under NZ$400
per month for it's replacement, the site may have to be slow for a while longer yet!