Fly Ryanair to Birmingham? Also you could see if there are flights to Manchester or East midlands (sometimes called Nottingham).
The A roads are OK from North Wales into the Midlands; Alton Towers is actually in the middle of nowhere and so hard to reach by public transprort or expensive via cab so having your own wheels from Ireland may be the best option.
Depends also when you arrive. If you are on a weekend or friday with a lot of tourist traffic then it could be a bit of a slog.
The other alternative would be on the english side. Get a train from Holyhead and go via Stoke or Manchester to Uttoxeter which is still about 20 mins away. And too many trains in the uK involve connections. One missed connection and you are kaput.