That's okay- can now give you some information/advice

Firstly, Ireland was part of the U.K. so no records were kept of people travelling between Ireland and England, Wales and Scotland.
Civil registration of births in Ireland started later than in England (1864) so there will be no birth certificates for the family. However, son Barney's Welsh birth certificate will list names of both parents so you will then get Ann's maiden name.
Was Barney the eldest child you've found? if so, once you know Ann's maiden name then look for marriage in Wales (registration of Catholic marriages in Ireland started 1864 so probably too late, registration of non-Catholic ones including Registry Office started 1845).