Many thanks for the advice.

I am now some six hours or more further down the road:  firewall disabled;
router reset to factory conditions with no security; antivirus off; windows
defender off.

I was expecting to find a port which needed to be opened for external access
but I am really not sure what I am looking for....

Not sure what all the HTTrack options mean so I have left them alone.  Action
left at "Update existing download" (as I have run the test more than once). 
Also "Do not use remote access connection".

I am completely lost on whether my problem is with the HTTrack settings; or
whether it is somewhere else.  All other programs (email, Internet Explorer
etc are working fine).  Could some IE settings be involved (cookies, pop-ups
etc?)  Any other ideas please?  Many thanks for being so patient.

Anthony