Can't see any reason the UK driving licence wouldn't be recognised (though yes, Irish tourism and Irish industry is going to be catastrophically hit by the impoverishment of Britain).

What will be affected is all those things that are part of being a member of a co-operative group of states - mainly trade agreements but also social agreements.

It's not possible to leave a trading co-operative but retain all the advantages agreed between members of such a co-operative. It's a bit like a student saying "I"m going to move out of this house where we all pool our money for groceries. I'm going to go and live in another place a few roads away, and use my own money to buy my groceries, pooled with the other people who live in that house - but I still want my share of free groceries from this house."