GITTOUTTAMAHPUBYOOSLAAGS!!
Now I have that off my chest...
If she works for the company, then they manage her employment contract & disciplinary issues, not you as her line manager, so you cannot ‘just sack her’.
Were both she & your OH not married/free agents & had entered into a relationship whilst working at the same location, one of them might be moved (at their request or given a hefty nudge) & this process would be documented in the employment contract, code of conduct or similar operational procedures. So moving one of them may not be a problem, you just need to find out company policy & get on with it. Given her length of service, I assume at that location, she will probably want stay put and you two would move. You cannot force her to move with you two staying as this starts of whiff of constructive dismissal. You & your OH might have to take the relocation in the chin. But that is for later.
So as a manager you need to get to grips with documented company policy, get your ducks in a row & refer this to your line manger/HR as this is all taking place on workplace premises, workplace time & will affect workplace conduct. It is not for you to solve on your own as there will procedures to cover this.
For your professional self-interest, the better you handle this the more likely you are to keep your job & possibly make a move that brings you some benefit - which is a conceptual bauble far away from the personal betrayal 
you are going through. Note the *caveats on speaking with your line manger/HR. You are obliged to inform them but before you do please give ACAS helpline a call as they may be able to bring you up on speed on workplace conduct & best practice, particularly in the pub trade. Also is there an association of licensed vituallers/pub management professionals with a helpline? I know nada about the pub trade but it seems that it may be contractually complicated.
Getting ducks in a row.
Are you going to present you & OH as a united team (knocked but firmly together) or are you uncertain of the future ? You need to know where you stand as a couple because you need to keep your professional mojo spinning. Your relationship may/may not survive this betrayal but why should your livelihood & what you have worked for be affected ?
Hmm, OH’s hand has been forced as her OH now knows. I think he had no intention of telling you & I am sorry to say you will have to drill through some deflecting crud to get to the truth. All the better for you to make the best decisions for you.
So, taken at face value, your OH is saying that a week or so before Christmas they went from everyday co-located colleagues to a frenzy of texting whilst they were apart for a few days, have just been kissing since & had a date a few days ago. Is that credible ?
Hmm, well I would not call that an emotional affair, I’d call that out as the early stages of a full-on affair - all that furtive, intensive yearning, betrayal, snogging stuff going on, not to mention the sneaking off together. Not slept together...yet. How did they get a recent ‘date’ past you & her OH ? Where did they go?
I think you may find (sorry to say) that this has been going on for longer & that the txting has been better deleted in the past. I wonder what prompted her OH to come across the message on her phone?
At this stage I think your OH is presenting a plausible low key story so everything can soon be glossed over. Emotional affair, pah just a bit of a flirtation. No need to fuss, will soon blow over. Blah, blah & thrice blah.
Your OH has to keep his job & his marriage, so he will minimise.
Sorry so much to say, time for stiff caffeine & creamy porridge.
Auntie Bear.