This was centuries before the UK. The Normans came to Ireland by invitation of Macmurphy, King of Leinster, to help him restore his power, in exchange for promises of territory. This barely counts as a conquest CB, but with certainty not as an invasion.
The original Tudors were Welsh.
Strangely, no 'English' people have ruled England since the latter Norman invasion.
Scotland and Wales often try to pretend they weren't part of the Empire and its horrors - in reality their were nobles/toffs/rich nobs from all across GB (at least) doing their part. Barely any of our ancestors were involved in any way other than servitude.
(My family are from both sides of the Anglo-Irish conflicts.)
The Common Travel Area's origins are in the the period 1923-1925[0], although it wasn't called that back then...