That household in 1851, in Nottingham St Nicholas, looks like this:
Jas Hallam, 51, born in Colveston, Leicestershire
Ann Hallam, wife, 55, born in Wollaton
- children:
Jas 19
Hannah 15
Mary 13
John 29, married, born Basford
- and John's family:
Catherine, wife, 31, born Nottingham
Thomas, 7
Eliza 5
Hannah 2
Mary Ann 6 mo
I think this is the same household in 1841 in St Mary:
James Hallam, 40
Ann Hallam, 45
John, 15
Thos, 15
James, 10
Ann, 14
Eliza, 8
Hannah, 6
Mary, 2
Catherine, 20
It looks like John's age has been rounded down from 19, and Catherine's from 21, in 1841 -- Catherine being John's wife and shown at the end of the list, after James and Ann's own children.
In 1841, the surname has been corrected by an Ancestry user, to Hallam from Hullum. That person can be contacted through the Ancestry system, to inquire about their connection, if any.
This marriage would fit that household:
June quarter 1841, Nottingham
John Hallam
Catherine Sherwood
If that were right, it would make you a genuine Sherwood from Nottingham.
In 1861, Catherine is in St Mary with children Eliza 15, Hannah 12 and Mary Ann 10, and also William, 4 months old, and is recorded as a widow. The 10 years between the two youngest children seems a little odd.