Thanks guys - I have now learnt the names of various technical guys at EE, HP and PC World. All, so far, have been about as much use as a chocolate fireguard despite some calls lasting almost four hours, and my learning about the vicissitudes of electronics far beyond my meagre grasp.
We have identified the fact that HP printers won't connect to extenders, that the EE router has a two band output - 2.4Ghz and 5.0Ghz - but that despite changing IP addresses, passwords, making strange incantations and threatening the machine with a fate worse than death - nothing happens.
So I have found the name of the EE CEO and emailed him personally, seeing as how everything worked fine until his brand new piece of kit ruined it all!