Thanks for the suggestions esmi – I just tried both but unfortunately no joy.
But I think I have figured out what the problem is…
I just asked my friend to visit the site with his iPhone and it worked no problem. So he suggested I visit another of my WordPress sites, and that gave me the same error message.
So I turned off WiFi on my phone and that stopped the error message.
It seems that it is something caused by our new Belkin wireless router, but that only affects WordPress sites (other sites seem to be unaffected).