My dh is Irish from Ireland and I have never heard of any Irish woman from Ireland being called Nana, but you could get a cute little pillow in Irish Lace, with Nana mongrammed in it or something along those lines. My Mil is called Granny and my fil is Grandad, but I don't think Nana is too bad. GL
Yes, randomly all the Italian women in our family are called Nana, but not the Irish ones (I am Irish/Italian so there is a fair number of both). So, I don't know if I can help you. All of our Nanaish stuff is Italian.
Yes, all of our Irish grandparents go by "grandma". I think you can change wording on cafepress though, if you want to get something that says "nana" on it.
Re: Irish Nana
Just to get the wheels turning, I googled Irish Grandma and the results were a lot of stores with Shirts, Mugs, etc.
Here's one from Cafe Press