Is it correct ? I don't currently have neither an irish residence nor an irish bank account yet
or I could say : I don't have an irish residence and an irish bank account yet
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I don't currently have neither an Irish residence nor an Irish bank account yet.
...is incorrect English, 'don't ... neither' would be a double negative. (Exception: if uttered in a rich Irish brogue, 'don't ... neither' becomes charmingly emphatic.)
I don't have an Irish residence and an Irish bank account yet
...also incorrect. It leaves the possibility that you might have one, but not the other.
I don't have an Irish residence or an Irish bank account yet.
Or, (by rearrangement, remove 2nd redundant 'Irish'):
I don't have an an Irish bank account or residence yet.
I have neither an Irish residence, nor an Irish bank account, yet.
Predominantly, when two negatives are used in a sentence, it usually just cancels each other out resulting in the sentence having a weak positive meaning (Except probably in cases where you're using double negatives to emphasize something). Since you've already mentioned 'You don't currently have' which indicates a negative there is no need to use 'neither... nor'.
You could re-write this sentence as 'I don't currently have an Irish residence or bank account' or 'I don't have an Irish residence or bank account yet'.
If you want to read more about double negatives, here are some links that might help,
Hope this helps!