I have Henry as born c.1820 in Carlisle son of John O'Neill (c. 1780, pure guess) and Catherine Shannon (c.1780 as aged 60 on 1841 census) I have a possible death for her in 1844 in Carlisle.
On the 1841 census there is a column for born in Scotland or Ireland and it looks like an I in it, so assume she was born in Ireland, but no way to tell where.
From the census I could see that Henry had a brother John and someone managed to look at RC Parish records in Carlisle for me and found a baptism for him in 1811 showing father John O'Neil mother Catherine Shannon. I think John must have died before the B,M,D records started in 1837 as I could not find anything for him.
Henry's sister Margaret shows father as John on her marriage.
Also from the 1841 census there is a Hugh O'Neill age 48 at the same address as Catherine, Henry etc. but as his name is O'Neill I have always assumed it is her brother-in-law, not brother. Quite a bit younger than her, or presumably her deceased husband, but families tended to be large and cover many years from first to last baby, so he could be a younger brother of John, looks like an I in the born in Scotland/Ireland/foreign parts column.
I have Henry's first 5 children born in Carlisle, they must have moved to Winlaton, near Gateshead around 1854-57.
Henry's wife was Margaret Gallagher, born in Scotland, appears to have had 7 children
John - 1843 (O'Neale)
Joseph - must have died young as another Joseph later, sound gruesome now but common then to reuse a name -1845-c.1846
Agnes Maron - quarter ending Dec. 1847 - from John's baptism a witness was Marion Gallagher (?sister of Margaret) so presume should be Marion non Maron
Joseph - q.e. March 1851, Carlisle
Robert - q.e. Dec. 1854, Carlisle
William - q.e. Sept. 1857, Gateshead
Catherine - c.1860-1951
Hope that helps a little, but I cannot tell you anything about anyone before John and Catherine.
If you are just starting to look at family history you might find freebmd.org.uk useful for births, marriages and deaths from q.e. Sept. 1837, as the name suggests it's free to use.
Family search from the Church of Latterday Saints if very useful too, though their new site seems to miss some of the records I found on their old one.
https://familysearch.org/searchI found out a lot by signing up for a free 14 day trial with Ancestry.co.uk then cancelling as it's quite expensive and I couldn't get anything more at the time.
GenesReunited is good as you can build your tree on it for free, you get 'matches' and 'hot matches' for people who might be in someone else's tree. You have to pay to see trees and make contact, but I got a lot of help when I did subscribe.
kind regards
Andrea
p.s. I was born in Northumberland, left at 21 and spent the next 29 years or so moving around the world with my husband's job. Back in the UK for 7 years, but near London.