Ok, problem solved.

I went to my local Apple store, here in Plymouth, UK. I showed them the phone and sure enough, I opened the compass app and it was as still as a mouldy moose.
Whatever the guy tried at the Genius Bar, nothing worked. So, without any bother they replaced my wife's 5s. I asked them to check the compass before we walked out of the store and it worked flawlessly.
When we got home my wife put her new phone in her protective case and thought nothing of it. An hour later she came into the living room and told me her compass isn't working!!!
Pretty p*ssed off with the whole episode, I laid my phone next to hers on the table and opened my compass app to try and compare settings side-by-side. My compass didn't work now either.
I then had that horrible feeling of stupidity wash over me. The case. Her protective case has a magnet in it to hold to protective flap open when using the phone. Gingerly, I took her phone out of the case and tried her compass. Bingo - working like a dream, it always did!!
Talk about not seeing the wood for the trees!!! It's ok, I'm over it now and my wife is back in love with her brand-new replacement shiny new 5s - which she didn't really need!!