Stan Carey has a post on ‘haitch’: a mainly Irish pronunciation of the name of the letter H. I have heard local pronunciations of other letter names in Ireland. In all cases it is the vowel in the name that is at variance.
Letter R with the NORTH vowel instead of the START vowel. This is widespread; I am unable to categorise the distribution other than to report that I have witnessed Irish people mocked by other Irish people for using either vowel. Perhaps “Cawstle Cawtholic” is involved. I notice RTE presenters in the last 10 years seem to have shifted more to the START vowel when saying the station’s name.
Letter A with the PALM vowel instead of the FACE vowel. This I associate with Irish-speakers, like my Mammy, for whom the A.B.C. is the /a: bi: si:/. Although Wikipedia notes distinct Irish-language names for all the letters, I’ve never heard them. Except for A, Irish speakers seem to use the English names: e.g. TG4 is called by its own presenters “[tiː dʒiː] ceathar”, not “[teː geː] ceathar” or “[tiː dʒiː] four”.
Letter Q with the STRUT vowel instead of the GOOSE vowel. This is a nonstandard Cork pronunciation. The same Corkonians pronounce the strong form of the preposition “to” with the same vowel, so it is never homophonous with “too” or “two”. Perhaps it is from the same motivation as “dubya” for “double-you”, mentioned below.
The other letter names mostly don’t vary.
The letter Z, of course, has different names in Britain and the US; Ireland tends to follow Britain, though I suspect that “zee”, like many US forms, has more currency in Ireland than England. I learned the alphabet rhyme with “…double-U, ex, wye and zee” but even then I used “zed” in other contexts.
The letter W is sometimes “Dubya”, as in the rustic Murkin nickname for George W. Bush, but in Ireland I’ve more often heard “dubbleya”, where only the final vowel is reduced, with the middle syllable preserved. Maybe this is the same as the process where final unstressed -o is reduced to schwa in nonstandard speech (so “tomorrow” rhymes with “Gomorrah”) though of course the final vowel in the name of W is the vowel of GOOSE, not GOAT. More likely it’s identification of the unstressed schwa with the stressed STRUT vowel. In the same way, there are Americans who have STRUT rather than FLEECE in the strong form of “the”; although in that word, in my experience, Irish speakers are more likely to have DRESS than STRUT.