In 1851 this may have been his family in Lemington, Newburn Hall, Northumberland:

First name(s)  Last name  Relationship  Marital status  Sex  Age  Birth year  Occupation  Birth place
George   Forster  Head      Married     Male     33  1818  Glass Founder  Newburn, Northumberland, England
Mary     Forster  Wife      Married     Female   32  1819  -  Newburn, Northumberland, England
William  Forster  Son            -      Male     14  1837  Glass Founders Son  Newburn, Northumberland
Jane     Forster  Daughter       -      Female   11  1840  Scholar  Newburn, Northumberland
Ann      Forster  Daughter       -   Female    9  1842  Scholar  Newburn, Northumberland
Mary     Forster  Daughter       -   Female    7  1844  Scholar  Newburn, Northumberland
Edward   Forster  Son             -      Male      3  1848  -  Newburn, Northumberland
Thomas   Forster  Son            -      Male      2  1849  -  Newburn, Northumberland
Jane     Porter   Mother In Law  Widow  Female   71  1780  -  Newburn, Northumberland

Ref:
Archive reference  HO107
Piece number       2413
Folio              307
Page               26
Combined:          HO107, piece 2413, folio 307, p. 26
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Going back a further 10 years to 1841, this looks like the same family at Lemmington, Newburn, Castle Ward, Northumberland:

First name(s)  Last name  Sex  Age  Birth year  Birth place
George   Forster  Male    20  1821  Northumberland
Mary     Forster  Female  20  1821  Northumberland,
William  Forster  Male     4  1837  Northumberland
Jane     Forster  Female   1  1840  Northumberland
Jane     Porter   Female  60  1781  Northumberland
Jane     Cook     Female   6  1835  Northumberland

Ref:
Archive reference  HO107
Piece number       823
Book number        10
Folio number       10
Page number        14
Combined:          HO107, piece 823, book 10, folio 10, p, 14
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Possibilities for his death, looking for a Stockton or M'bro' registration with a birth in 1837:

Name:                    Age at Death (in years):  
FORSTER, WILLIAM     51  
GRO Reference: 1888  D Quarter in STOCKTON  Volume 10A  Page 26

Which is the only with those criteria.

Was in (a) the 1881 census or (b) the 1891 census?

Yes he was definitely around in 1881 living at 103 Haverton Hill, Stockton:

First name(s)  Last name  Relationship  Marital status  Sex  Age  Birth year  Occupation  Birth place
William     Forster  Head      Married  Male    44  1837  Furnace Foreman (Iron) (Gl)  Northumberland
Elizabeth   Forster  Wife      Married  Female  44  1837  -  Dringhouses, Yorkshire
George      Forster  Son       Single   Male    20  1861  Fitter Apprentice  Billingham, Durham
Annie       Forster  Daughter  Single   Female  18  1863  -  Billingham, Durham
Mary Jane   Forster  Daughter  Single   Female  15  1866  Pupil Teacher (Sm)  Billingham, Durham
Elizabeth   Forster  Daughter  Single   Female  13  1868  Scholar  Billingham, Durham
William E   Forster  Son       Single   Male     5  1876  Scholar  Billingham, Durham
Bertram     Forster  Son       Single   Male     1  1880  -  Billingham, Durham

Ref:
Archive reference  RG11
Piece number       4893
Folio              122
Page               40
Combined:          RG11, piece 4893, folio 122, page 40

And he was not to be found in 1891.  Sounds like QED: he died in 1888.
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Thier marriage, almost certainly a church record:

First name(s)           William
Last name               Forster
Age                     22
Marriage year           1859
Marriage date           24 Oct 1859
Marriage place          Billingham
Spouse's first name(s)  Elizabeth
Spouse's last name      Hardcastle
Spouse's age            23
Spouse's birth year     1836
Spouse's birth date     1836
County                  Durham
Country                 England
Record set              England Marriages 1538-1973
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