I have determined the cause, and a work-around, but I am still surprised by the way it works.
The DNS forwarding was by 'mask', so when the user logged on to www.MyIWPFile.co.uk they saw that address in their browser address bar. When the forwarding was changed to be by 301 permanent (or 302 temporary) then it works.
I can only assume that every time a subsequent call makes a reference to 123.456.789.0 then it re-refers back again to the DNS - but that seems really strange and unnecessary - almost like a reverse lookup. And I can't see why right-click should make a difference to the functionality.