I imported a car from the south into Northern Ireland and it was a gift to do, drive it home, you won't get a tax book as the southern set up isn't like the uk set up, the old owner will post it away for you , once you get home Mot the car on chassis plate, fill in whatever form ( can't remember ) and send new mot certificate and it will register the car to the uk, pay the £80 fee and that gets you a tax book and a number for your plate, once you have that get number plates made up and tax the car, it's as easy as that
Now importing a car from U.K. To Southern Ireland is a totally different ball game costing thousands of euro
Also to point out the Irish police are like the coast guard useless and to busy doing other things , the car might still be taxed and Nct ( Irish mot ) get yourself insurance and drive to the boat, once in the uk the uk police will have no idea if its taxes or motd so you should be fine even if it doesn't have any at all, as long as your insured
Remember it will be on Irish number plates and won't even register with the U.K. Police, or get the lend of trade plates lol