Well, along with pretty much everyone else from Ireland, I have studied the modern language for 14 years. That being said, it is not used for daily conversation outside of very limited areas (the Gaelteacht). Back at the time of the Anglo-Saxon invasions, the language here would have been Old Irish - which was quite a bit more arcane and complex (well - at least according to the little I have seen of it), though certainly recognisable.
In Britain, Brythonic would have been spoken - which would be much closer to modern day Welsh (Britain was likely settled through France, and Ireland through Spain at perhaps 350 BC- the Celtic language had diverged earlier yet).
Trace it back, and you get earlier peoples in the Isles, conquered or assimilated. These had their own cultures, languages and monuments (Newgrange is over 5 thousand years old, and is older than the Pyramids). These too are lost - remembered only as Formorians, Firbolg, or Milesians.
Now the Welsh were these Britons existing in a land yet unconquered by the Saxons - indeed their very country is named by the Saxons for it (Waelas - foreigner, often a Celtic (sometimes Roman) epithet. Wallachia and Vlad have the same root). As was Cornwall (Kern Waelas - Foreigners of the Horn).
And yes, England experienced many many waves of conquest, as evidenced by our mongrel tongue. Perhaps the most remarkable part is that their relative integrity in the recent Millenium or so. The same, unfortunately, could not be said of Ireland - although in earlier years we got off lightly. Indeed, post Roman times, we were raiders too: Ever wondered why Scottish is just a version of Irish?.