A walking sick or cane is legal everywhere, so far as I know, whether made of Blackthorn or any other wood.
I've traveled all over Ireland and elsewhere in Europe with walking sticks, and never had a word from the local constabulary.
The traditional Irish shillelagh was sometimes quite a bit shorter than a cane, and might well be viewed in the same (legal) light as a cudgel or bludgeon, accordingly, but if it was of such a length that you were using it as a walking stick or staff, there should be no problem. Just be sure you don't pour molten lead down holes drilled into the head of the stick, as they used to do a century ago!
Currently you should be especially welcome if you're a tourist in Ireland; they're in a recession, and glad for tourist dollars!