To expand on @JanMurphy's first method, there's a set of rules that genealogists use when they get stuck:
- If you can't go backwards, go forwards
- If you can't go forwards, go sideways
- If you can't go sideways, go to the pub
You've been unable to work back from Norah's later life, so try another approach. Start with something you do know. You have a candidate for Norah's birth. That birth certificate describes a family: James, Ellen and Ann Norah O'Brien, living at 12 Church Way, Somers Town, St Pancras. The question is, is that your family or another? So the goal is to research that family, and see if their Ann Norah can be confirmed or eliminated as your Norah.
From that certificate, I would assume that James O'Brien married Ellen O'Donnell some time before 1860. A quick search brings up a good candidate on Ancestry in their "London, England, Church of England Marriages and Banns, 1754-1932" record set:
Married on June 05 1853 at St Pancras Church, St Pancras were:
James Joseph O'Brien Full Age Bachelor French Polisher Father John O'Brien
Ellen O'Donnell Full Age Spinster -- Father Jeremiah O'Donnell
The names are a good match, as is James' occupation, and the date and location are a decent fit for this family. One witness is Matthew O'Brien, who might become interesting at some point. And we gain a middle name for James, and the names of both fathers. Both are of "full age", which would normally be 21 then, so we might assume that both were born earlier than 1832 or so. Or perhaps a year or two later, these things weren't always exact, or checked.
Can we find them in the 1861 census? Ann Norah would be less than a year old. This is a little trickier, but there is a decent candidate in Somers Town, St Pancras (RG9; Piece 110; Folio 43; Page 17 on Ancestry)
Here we have, I think, spread over two pages:
James Brien Head 27 Driver of Hackney Carriage Middlesex St Andrews
Ellen Brien Wife 29 -"- Wife Middlesex St Giles
Mary Brien Daughter 7 Scholar Middlesex St Pancras
William Brien Son 3 Scholar Ditto
Hannah Brien Daughter 7m Ditto
The writing is very unclear, and James' occupation could also be something like "Donor of Wedding Carnage" but I think the more likely options are as transcribed above. Oddly, the census taker has written in an uninhabited property between James and Ellen and the children. I think he's put that out of sequence by mistake.
There are several problems when we compare this family to the information we already have. The surname is wrong - Brien instead of O'Brien. James' occupation is different from that on the birth certificate just a few months earlier. There's no Ann Norah.
But look at the address (it's clearer on the previous page): 12 Churchway, Somers Town, St Pancras. It's the same address as James and Ellen on the birth certificate. I doubt that's just coincidence. And people do change their jobs quite abruptly at times. And Hannah is essentially a version of Ann (or could even be a mishearing of Ann Norah). She is the right age for Ann Norah. I think it is plausibly the same family. (That's not to say you shouldn't look for a better match, of course!)
This gives us more to work with, anyway. What does the new GRO births index have for O'Brien, with MMN O'Donnell? Sadly, Soundex seems to think that O'Anything names are pretty much interchangeable, so a fuzzy search on those surnames brings up a lot of chaff. But confining our results to the London area from 1853-1861, we get:
Margaret O'Brian Q3/1854 Bermondsey
Mary O'Brien Q2/1854 Saint Pancras
Jeremiah O'Brien Q1/1854 Strand
Matthew O'Brien Q3/1855 Strand
Ann Norah O'Brien Q4/1860 Pancras
John O'Brien Q2/1860 Camberwell
There's clearly more than one O'Brien/O'Donnell family in London at the time, which is not terribly surprising. But there's a Mary, born in the right year in the right place, and Ann Norah of course. No William. Perhaps the surname 'went wrong' again. If we repeat the search for Brien/O'Donnell, we get one more result:
William Brien Q2/1858 Saint Pancras
So we have candidates for Mary and William as well as Ann Norah/Hannah, all registered at (Saint) Pancras in years that match the census ages. This looks like a five-membered family group, ripe for further research. You might want to expand the search to look for later births, too.
We've also seen that we can't rely on specific names and dates being exactly correct on any given record. And you can't rely on Soundex to pick up all possible variants. That ambiguity isn't going to go away. Mis-spellings and mis-tellings are part and parcel of family research, and you have to accommodate them. Often, it's not really possible to "prove" that you have the right person, but you can try to use the weight of evidence to put the matter beyond reasonable doubt.
The challenge you have now is to follow that family forward. Find out what happened to each of them, if you can. You might find, for example, that Ann Norah/Hannah ends up on a path that means she cannot possibly be your Norah (e.g. dies young, or marries, or emigrates, or…) in which case you can eliminate this family from your research. If you can't rule her out, then keep on plugging away until the weight of evidence one way or the other is irresistible. And then go to the pub.