This is almost our family. My children were born in Belfast to me an American by birth with an Irish passport by post nuptial declaration and a Belfast born (Irish ) father. They hold all 3 citizenships. I have US and Irish citizenship and eligible for a UK passport as well. I haven't pursued the UK naturalisation because I dont need it since we are in the EU and the Belfast Agreement says we can be Irish, British or both. There is £900 fee plus English and citizenship tests and an oath of alleigence to the Queen. I have a friend whose kids had the 4 mum and dad's countries of birth,Irish and British at birth.