Do as much research you can in the US, you may find something that leads you to an actual location in Ireland which will be very helpful. You will find that there may be thousands of people by the name of Nicholas Connelly. In the north the earliest census are parts of the 1831 Census, birth, marriage, death registrations were not required until the mid 1840's. You may find church records earlier however you will need to know the location.
The Catholic Parish Registers for Ireland are here and are free but you must know the parish and there is no alphabetical searching, you must go through the parish records by date, if available.
Be careful as there are many sources that are free, that you will be charged for by some sites.
If you can find the location in the informaiton in the U.S. please get back to us, as that will open up many doors.
Good luck in your search, I have been researching my family in Ireland for the past 25 years and have made many trips over searching for them.
North records are usually separate and apart from the South/Republic of Ireland, the North is part of the UK and the South is it's own Republic.