There was also another child born to John Buxton's 1st marriage to Susanna: Jane Buxton Bapt:9th Oct/1843 Haxey- Died: 22nd Aug/1847 Haxey, residence Westwood. Age 3
As to Clara Jane Buxton born 1849 one assumes this would have been Susanna's child, since Susanna died in 1850.
As to the birth registry of Clara Jane Buxton, I have not found one. However, there is a Sarah Ann Buxton registered 1850 Dec.qtr. District: Lincoln (source Free BMD).
I know this is somewhat late to register an 1849 birth but John had lost his daughter in August 1847, his wife in May 1850, and was about to make a move to Sheffield, a lot to handle for sure.
I am wondering if John or whoever registered the above birth was very distraught,confused and in a hurry and by mistake, gave the other daughter's name Sarah Ann(mother Susanna) who was born/bapt 1841 at Haxey.
John and his two children moved lock stock and barrel to Sheffield. married Elizabeth Wheatcroft, all before the 1851 census was taken. A lot going on here..!!
GRO Marriages JUN 1869
BELL James
BUXTON Clara Jane
Carlisle 10b 642
A tree at Ancestry and censuses after this marriage say this Clara Jane was born in Sheffield (one census says Carlisle). She is about the right age. I know the Clara Jane we are looking for was (probably) born in Lincolnshire but she would have thought of Sheffield as her home town.
I don't know how she would have ended up in Cumberland and I can't find a record of the marriage online that gives details of her father and his occupation.
The BELL family had roots in Scotland. Maybe that's where the BUXTONs disappeared to?!
If this does turn out to be the right one there is a photograph of her in the Ancestry tree. She died in 1923.