Yes, this Brexit thing is a bloody nightmare.
I live in the UK, hold an EASA ATPL issued by the UK CAA. So far so good.
I fly a Portuguese reg aircraft, and do my recurrents in the US at Flight Safety.
Many of my colleagues have UK ATPLs as the examiners in the US have authority for a few NAAs (UK, Belgian, French, German...)
I don't know if they have Irish rights. :/
Yes, these questions are in the office....
It's just a bloody mess.