Hallo, Alicef
On the 1881 census there is one James Unwin, son of William and Hannah:
RG11/0499, Bromley, folio 128 page 30
91(or 41) Bowlem(?) Lane, Bromley St Leonard, London
William Unwin, head, mar, 35, Labourer, Cambs - Ely
Hannah, wife, mar, 27, Ireland - Cork
Mary A, daur, 10, Scholar, Ireland - Cork
James, son, 7, Cheshire - Chester
William J, son, 3, Lancashire - Liverpool
Delilah, daur, 1mo, Middx - Bow
There are only 2 other James Unwin, son of William:
RG11/4691, Doncaster, Yorks, folio 126 p6
James Unwin age 11, son of William age 50 and Ann age 47
RG11/4462, Bradford, Yorks, folio 99 p19
James Unwin age 7, son of William age 47 and Maria age 37
also:
RG11/3496, Macclesfield, Cheshire, folio 96 p9
James Unwin age 7, son of Harriet age 46 - married, but husband not in the household
It is a reasonable bet that the Bromley Unwins are the ones you are looking for.
On the 1851 England census there is:
HO107/1764, Ely, folio 336 p29
both sides of Newnham to West Barns Gate
Robert Unwin, head, m, 36, Lab, Cambs - Ditton
Mary Ann Unwin, wife, m, 35, Domestic, Cambs - Ely
James Unwin, son, u, 16, Lab, Cambs - Ditton
John Unwin, son, 8, Scholar, Cambs - Ely
Willm Unwin, son, 6, Scholar, Cambs - Ely
Mary Ann Unwin, daur, 1, Cambs - Ely
I can't see the Unwin family on the 1861 or 1871 England censuses.
Hope this helps
Philip