Hi Friesland -- as you say there are no 'migration' records from Ireland to England, as we were all one country back in the day.
It may be useful for you to now put your queries separately on the Irish Board. By that, I mean your Hickey and Carroll families as 2 separate queries.
With Thomas Carroll - at least you have a father's name, so someone on the Irish Board (and there are some Irish 'whizzes' on there) - may be able to find Thomas's likely birth.
If you do proceed with that, you are better putting a link to this earlier correspondence, so that people don't just go over the same ground.
Have a think about it - I am out of my depth now, having exhausted my meagre knowledge of the frustrating Irish research!
(I have an Irish grandmother AND my husband was Irish - so believe me, I do understand the difficulties).