Update - having switched the PC off overnight, it now appears to be working normally! (This is being sent from the PC).

because it is working, but

for what was wrong in the first place!
I HAD tried a couple of restarts of the PC (as well as the router) before calling for help. The only combination I hadn't tried is powering them both off together (if that has any relevance?)
Maybe the cable didn't work on the laptop because it was the wrong port? But it was the only hole in it that fitted the type of cable ending (and has a networky type of icon next to it).
Computers are brilliant, when they work...
Thanks for all your input, guys.