This is intended as a supplement to the previous answer.
The Indexing Problem
Whenever you have a chance to search "the same" data at different websites, it often pays off to search the data on all available sites. You can't always tell if the sites are working from the same computer index, or whether each site has made their own index. Search engines differ, and a search result which eludes you on one site can show up on another.
I highly recommend anyone having difficulty with finding someone in the GRO indexes watch the 2018 Rootstech presentation Civil Registration Indexes of England and Wales by Audrey Collins. Collins is the co-author (along with David Annal) of Birth, Marriage, and Death Records: A Guide for Family Historians; she wrote the three chapters on Civil Registration. Collins discusses the whole process of how the GRO indexes came into being and highlights the parts of the process that cause opportunities for errors to creep in. The talk begins with the marriage indices and how they are often more deficient that the indices for births and deaths.
The process of generating the index volumes involved several steps, including transcribing the quarterly returns sent in to the GRO. Any time you have records which are copied, there is the possibility for errors to creep in, because of the clerk not being able to read the handwriting, a difficulty in sorting the slips, accidentally skipping entries when copying, etc.
From 1837-1865, the indexes were copied onto parchment index volumes. The work was task work (clerks were paid by the number of entries) and while senior clerks did check some of the entries, even the best clerks made errors.
Collins recommends always going to FreeBMD first because they are free, and because of the high quality of the indexing and the flexibility of the search on the site. FreeBMD indexers work from the microfilm and microfiche. Ancestry's copy of the GRO BMD indexes is in two databases, one up to 1915, and one from 1916 onwards. The earlier BMD indexes (to 1915) were obtained from FreeBMD. Each of these databases is likely to be a snapshot of FreeBMD taken at one point in time. They aren't likely to have any corrections made later or to give you the PostEms left by FreeBMD users.
For their online searchable indexes, the GRO went back to the original quarterly returns. Even so, the index has errors. The age at death can sometimes report the ages at death which were reported in weeks or months as years, because the indexers didn't follow the instructions to record those ages as zero. The search engine doesn't force you to use exact search -- you can search for similar-sounding names -- but they don't tell you what methods each one of those choices are using.
Findmypast is adding in to their version of the birth index the mother's maiden name information, and there you can search by the mother's maiden name alone, a search you can't do at the GRO's site.
When using any site, it always pays to run wide searches looking for irregularities in coverage. Recently in the Lost Cousins newsletter, Peter Calver wrote several articles about large deficits in the GRO's online searchable index. Who Do You Think You Are magazine did a recap in their news article: Home Office responds after genealogy blogger finds records are missing from GRO indexes by Rosemary Collins, posted on 15 November 2018.
The Geography Problem and the Date Problems
When you're using any record as a guide to finding another record, it helps to think about evidence analysis. See QuickLesson 17: The Evidence Analysis Process Map at Elizabeth Shown Mills' website Evidence Explained. Consider that:
We don't know who gave the information to the enumerator.
None of us have primary knowledge of our own birth date or place of birth; we were there, but we weren't in the position to read a calendar or map. Someone else had to tell us when and where we were born: this is secondary information.
The 1901 and 1911 Census are rather far removed in time from the birth. Records which are closer in time to the birth may be more accurate (but we should also take into account Dr. Thomas W. Jones' advice about the Perils of Source Snobbery).
Over the course of an individual's life, you may see many different places reported for their place of birth. It helps to use maps and gazetteers while you are searching, such as GENUKI's gazetteer, British History Online's copies of Lewis' Topographical Dictionaries, or a site like A Vision of Britain.
An example from my own research: I was unable to find my father's entries in the US Federal Census immediately after his birth because I didn't remember what town he grew up in. I remembered a big city which was near his hometown, where he lived later on in life. Once my older brother reminded me where the family had lived, and I used wildcards instead of searching for the surname the way my family usually spelled it, I was able to locate the right census household quickly. People saying they were "from" a bigger locality instead of the smaller towns which are nearby is a very common problem, and especially so if the person giving the information is a different person in the household. The conversation goes like this:
- child: "Where are you from?
- parent: (name of small town)
- child: "Where's that?"
- parent: "It's near (name of bigger town)"
It doesn't take long before the family only remembers the name of the bigger town and not the name of the actual hometown. Now consider what happens when the town is near a county boundary. The birthplace shown on a census record may not even be in the right county.
Similarly, relying too much on the ages in the census and on the calculated years of birth reported on websites can cause us to miss a registration because we were looking in the wrong date range.
Study all of your 2x great-grandmother's family and look at the entire set of records, not just from her own records, but those of her siblings as well. Establish where the family might have been living when she was born by looking at other records before you try to locate the birth registration. Comparing several census records can show you the family's migration pattern if they didn't stay in the same place. Look for any records which can help you locate the family at particular points in time and map them all out.
You may have to use FAN Club research (friends, associates, neighbors) to find the answer. See Elizabeth Shown Mills' QuickLesson 11: Identity Problems & the FAN Principle for a worked case study.