Each state has different divorce laws. Google divorce laws in your state. It tells you how long it takes for uncontested vs. contested. There is a difference between not signing the papers, and not responding to your petition for divorce.
Educate yourself..... it helps get rid of all the "what ifs".
The location for the OP says Dublin, so state laws (guessing you mean America) won't help her if that's accurate.
I read up a little on Irish divorces and it seems that it takes 4 years of living separately before the proceedings can actually begin? Is that right? That just seems crazy to me, and this is coming from a guy whose wife recently moved out and is supposed to file any day now for a divorce I don't want.
If you are living in Dublin and that is the case for you, I'm really sorry, that's a ridiculous amount of time to wait.
But in terms of how to deal with him, I'd call, text, see him face to face or whatever you're comfortable with and just say there is no chance at reconciliation. You don't have to be mean about it, but be firm and then tell him that after that conversation you won't reply to any communication (you can set some parameters where you're okay with it if you like—i.e. emergencies, something for the divorce etc.).
After that, stay firm. Just out of curiosity, how much communication is taking place between the two of you now? Understand that he will equate any communication to hope that the marriage can be saved. Even if it's you telling him over and over it's done, the constant engagement will send his scrabbled brain the message that as long as you're talking to him, there's a chance something can change your mind.
Hope it gets better for you.