It's a decade on since the original post, and I'm still searching for Frances. Honestly until recently I was not much further on despite a number of attempts. I recently made a little breakthrough which I thought I'd share for anyone interested.
At the beginning there was the possible Frances Johnson of the right age and birth place recorded on the 1861 census in Sneinton Asylum.
I did recently have chance to check the Sneinton Asylum admission index (record only opened to the public last year as it was closed for 100 years, and the last entry was in 1919).
This record showed that Frances Johnson was admitted to Sneinton Asylum on 26 Nov 1841. Noted as a widow and age 38 at admission. The index notes that she died on 19 Nov 1863.
Now, finding other record of this death record was a struggle, as there were no Frances Johnsons in the GRO index who died in Dec quarter 1863 of the right age or place.
Looking in burial records I did come across this burial at Nottingham St Mary's, which seems about the right date:
23 Nov 1863, Frances Sharp, age 60, of Asylum
There is a Frances Sharpe in the GRO death index which ties with this burial. I just ordered this death certificate to see if it contains any more useful information.
On the 1851 census, there is a "Frances S." (no surname given), who was a widow, age 48, born Upton.
I suspect Frances Sharp and Frances Johnson are the same person. My Frances did have a son born in 1824 in Upton, Notts, which might explain why that 1851 census entry showed Upton as her birthplace.
The question is whether I could have found my Frances at long last, or whether I have been barking up the wrong tree for so many years. The other question, if this is the right Frances, why using the surname Johnson sometimes and Sharpe others?
I'll post the death certificate info when I receive it.
Alexander