Long, boring comment now follows.
As I suggested before, there are just three possible births in the index which match Ms Brown/Hall:
Dec 1852 Sarah Ann Brown Basford 7b 109
Mar 1853 Sarah Ann Brown Belper 7b 407
March 1853 Sarah Hall Belper 7b 416
We also know, from the Family Records Centre index for marriages, the Sarah Ann in question was married in her “adopted” name:
March 1873 Joseph Cresswell Belper 7b 707
March 1873 Sarah Ann Hall Belper 7b 707
Indeed, it may be worth knowing if she had a natural mother still living.
When we start looking at the 1861 census, I assume we can ignore the Dec 1852 birth in the Basford district, as out of area. That leaves just two possibilities. What comes next is taken from my reading of the 1861 census: for reasons that are explicit from my earlier, censored and mangled contribution to this thread, I am having to be circumspect (and unnecessarily wordy).
Sarah Hall, presumably she whose birth is in the index for the March (first) quarter of 1853 (reference 7b 416, cited previously), would appear to be the daughter of William and Sarah Hall. On census-night, 7 April 1861, she is safely tucked-up at home with her three sisters and four brothers. Her birth-place is declared to be Swanwick.
The same night, there is a slightly-older Sarah Ann Brown, Belper-born, aged 9, in the household (12 Long Row, Belper) of her parents. Her parents are Jesse Brown (age 32, “Hosiery Warehouse Clerk”) and Orah Brown (age 37: b. Belper). Also in the house is Martha Brown (aged 18: “sister-in-law”, b Belper) and another daughter, Florence Brown (aged 2: b Belper). There are lodgers: Thomas Flint (aged 22: b Notts) and his wife Amelia Flint (aged 17: b Blakesley, Lancs).
There are two possibles for "Sarah A Hall" in the 1861 Census:
At 66 Laund Hill, Belper, we find the family of John Hall (aged 38, Labourer; b. Belper) and Mary Hall (aged 33, “wife”; b. Belper). They have three daughters: Sarah A Hall (aged 8; b. Belper); Eliza Hall (aged 2; b. Belper) and Pheoby Hall (aged 4 months; b. Belper). Hmm: not likely.
But all is not lost!
At Capel Cottage [which seems to be on or off the Burton Road], in St Werburgh’s, Derby, there lives James Bennett (aged 53, Labourer: b. Marylebone, London) and his wife, Jane Bennett (aged 42, b. Chadderton, Derby). They have three children: Georgeania (aged 15, “House servant”, b. Ripley — “Surrey” has been superscribed in a different hand, almost a scribble); Joseph (aged 8, Scholar, b. Summerhill, “ditto” – presumably ditto to “Derby” in entry for Jane Bennett) and Paul (Son, aged 6, Scholar, ditto [to Summerhill]).
There is something very fishy about the “Surrey” addition. The original enumerator did not feel any need to include it in the entry, even though the county names have been scrupulously entered previously. I believe Summerhill may be a farmstead near Monyash, Derbys, by the way.
Now: roll of drums; crash of cymbals…
Also in the Bennett household for census-night is Sarah A Hall (Visitor, aged 8, Scholar, b. Ripley “ditto”, – presumably ditto to “Derby” in entry for Jane Bennett). The handwriting for the “ditto” is that of the original enumerator, and the various “dittos” are, to my eye, definitely for Derbyshire. The modern transcriber (i.e. the ancestry.com apparatchik) is in barbed-wire canoe territory here: no way can that ditto refer to the superscribed “Surrey”.
Q.E.D.