You really need Hanora's surname to progress, that's why I was focusing on any children born in England...
As you have seen the names are quite common even in England, so would be even more common in Ireland, especially around the south west where Regan is most prevalent. In addition given the early dates the availability of records will be a factor - the average starting date for Cathic records in rural areas was about 1830 and some parish records start even later, so locating and confirming baptisms for Jeremiah or Hanora with any degree of certainty is a problem.
If Hanora's surname could be located, e.g. via birth or baptism of later children in England, then you could search for earlier births and baptisms in England or Ireland and possibly at least establish a parish in Ireland for the children which would likely be where the couple originated.
A quick search or the available Catholic records for a possible baptism for a Denis Regan (a slightly less common first-name) in Ireland 1844 +-5 years with father Jeremiah or similar shows about 2 results, both in Co. Cork parishes in 1839 and 1849, however neither is a good match as parents are Jeremiah and Margaret and Jeremiah and Mary. It's possible that the entry for his baptism is there somewhere but faded or difficult to read, but checking the individual registers in detail would be next to impossible as Co. Cork is the largest county in Ireland with something like 100 Catholic parishes...
A search for the far more common name John Regan c1832 with father Jeremiah finds about 13 matches, again all in Co. Cork parishes, one of the mother's on these is difficult to read, but looks could possibly be an abbreviation for Johanna, surname looks like Taylor ? Might be worth considering.....
The record is of a John Regan baptised on the 5th May 1830, father Jerh., the parish is Bandon RC, which was based in the town of the same name... this is a link to the register page with this baptism (see left page abt half way down), and I've also attached an image extract of the baptism in question..
