OK, after the hiatus whilst I tried to get to the bottom of the problem, the site’s is back up. I reverted to the earlier Silesia theme while I played with the Headway 2 theme using the Theme Test Drive plugin.
I couldn’t get to the bottom of the problem so I replaced the Headway theme with one from another site; which is broadly similar. Unfortunately, it seems that something deeper was broken with Headway because this approach didn’t work.
To cut a long story short, I deleted Headway 2 and installed fresh Headway 3 Base. Again, I customised this in Theme Test Drive so that the live site wasn’t affected by me building a new theme in the background.
Headway 3 is significantly different to Headway 2, so it took a bit longer. It’s not right, but it’s close to where it was: enough to go live anyway.
I’ll tweek it over the next few days.