Hi Evie
Thanks for that, your help is much appreciated!
No, that is not the right family in Westmorland!

Jane was pretty consistent giving her birthplace as North Sunderland, Northumberland (1851, 1861, 1891) but in 1871 it is just Northumberland and in 1881 it was Sunderland, Durham (Jane and the younger children were living in Horsley, Ovingham and Robert and eldest dau Dorothy were visiting relatives in Jarrow).
What varied was her age - born anywhere from 1830-1835. Her husband Robert Joseph DIXON was several years younger (born 1838 in Longframlington, s/o John DIXON and Dorothy nee REED).
Jane and Robert DIXON's children were:
Dorothy born 1858 Annitsford, Northumberland (my ancestor)
John born 1860 Annitsford
William born 1862 Annitsford
Mary Jane born abt 1864 Annitsford
Isabel born abt 1866 Annitsford
Robert born abt 1867 Annitsford
Nicholas born 1871 Ryhope, Durham
Margaret Ann born 1875 Horsley, Northumberland
John Joseph born 1877 Horsley
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It is possible it is the wrong family in Wooler with the births and christenings for Jane 1830, James 1832 and Margaret 1834 to parents William and Jane but there is no other match for Jane in the right time period and general location. The IGI is not complete, of course, which is why I put the query in the hope of either confirming or eliminating this family. I've now heard from someone who has access to the records for the Wooler Presbyterian Church and she says the children of William and Jane were born at Lilburn Grange, Wooler.
I obtained the various census records 1851-1891 some time ago but do not now have a subscription to A ncestry and had then put the family on the back burner. I only had a "best guess" at that time that Jane's surname was AINSLEY/AYNSLEY as she was the
only one who came up in 1851 (aged 19, working as a servant in Cartington, Nbl) as having been born in North Sunderland. (I think the 1851 index at A ncestry has her age as 29 but the image looks like 19.)
It was only this weekend that I decided to revisit FreeBMD and see if the marriage to Robert DIXON had been transcribed (marriage records for the late 1850's have been incomplete) and found Jane AYNSLEY on the same page as Robert DIXON marrying just a few months before Dorothy's birth (cannot buy certificates for now).
When I tried to check there were
no other AINSLEY's or variations coming up on the censuses in the time period of Jane's approximate birth (to be her siblings) so began to wonder if she had got it wrong and decided to take a look at the Wooler family ~ turned out not to be as easy as it first appeared!!

I would say, giving the children's names, that William is a strong possibility for Jane's Dad's name but not sure about her Mum as Isabella could also be a possibility rather than Jane.
[The name Nicholas comes from Robert Joseph DIXON's maternal Grandfather Nicholas REED.]
However, even knowing for sure if William is the name of Jane's Dad is not really going to help at the moment as there appears to be only the one match in 1841 that you found for Jane with Jane SMITH (widow) in Newcastle.
They may not have been in Newcastle long though as checking FreeBMD for a birth for 6-month old Isabella SMITH there is only one possibility in Belford registration district (which includes North Sunderland). Isabella's christening is not on the IGI either. Trying to follow up on the SMITH family in case there is a connection has also drawn blanks.
Burials for the Wooler family would be nice to rule them out completely as it appears that only James (1832) survived and he was in the workhouse as you found in 1841.
There appear to have been no other AINSLEY/AYNSLEY children in 1851 born circa Jane's birth so was she too the only survivor of her family?
So, that's the current situation in a [rather large] nutshell!

Thanks again for your interest and findings!
Lynne x
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God is good!