Wow, brilliant detective work, thank you so much. I've been searching for days. I'm sure that's the answer!
Sorry I can't find Jane Elizabeth Jackman/Williamson on 1861 census but this looks to be Jane Williamson who is widowed
https://www.familysearch.org/ark:/61903/1:1:M7NS-9F6I also noticed on the 1851 census for Elizabeth Jackman with her mother Ann Dodsworth
https://www.familysearch.org/ark:/61903/1:1:SP9N-VT6 the address is the same as for John, Jane & Jane E Williamson, Queen Street Ripon
ADDEDDid one last try have not found Jane Elizabeth Jackman on 1861 census
You may already have these details as you mention she is not with her mother on 1851/1861 censuses
1861 census Christopher Dodsworth age is living with his son in law
Thomas Bannister and his wife
Elizabeth born 1828 Leeds and their 5 children
https://www.familysearch.org/ark:/61903/1:1:M71N-SH5BANNISTER, SARAH ELIZABETH
mmn JACKMAN 1852 D Quarter in EASINGWOLD Volume 09D Page 294
Thomas & Elizabeth Bannister plus 8 children
1871c
https://www.familysearch.org/ark:/61903/1:1:VBHV-S3SBANNISTER, EDWARD JACKMAN
mmn JACKMAN GRO Reference: 1864 M Quarter in GREAT OUSEBURN Volume 09A Page 87
other birth entries on GRO indexes showing mmn Jackman
Dec 1851 Ripon
Thomas Bannister
Female name on same page
Elizabeth JACKMAN
Do you know Elizabeth Jackman's father's name & his occupation?
The marriage certificate of Jane Jackman to John Williamson should give you
her father's name and occupation and hopefully it matches that of Elizabeth Jackman's marriage