To update/close this one.
It's now been resolved, though not solved.
Before my son was able to get over to have a second look at his in-laws printer they called in a local 'PC Technician' to have a look.
"He found there was a lot of ink in the bottom of the unit which was causing problems. Plus the printer is approx 10 years old and used regularly.
They now have a brand new HP printer now, which has both wired and wireless capability. They’re running it wired as they have no wireless facility in the PC."
So they're happy and they can now print OK, but clearly we'll never know the root cause of the problem. Clearly the excess ink in the bottom of the unit wouldn't have helped but certainly wouldn't have resulted in (apparently) different behaviour between different users.
I say apparently
as there is a hint from reading between the lines of conversations that there was no controlled test, e.g. a simple one such as printing a simple B&W document under each user, similarly a colour one. So perhaps the problem was more one of confused understanding.
As I'm sure people can appreciate it's frustrating when one is ~250 miles away and having to diagnose from confused explanations.
However, despite all that, a big thank you to all who took the time and trouble to make the assorted suggestions, they're all helpful to store away for the future