Hi I cannot assist with help in this region around the time frame you are hoping for, however I came across this entry for an Enoch Spencer born c.1853
Dwelling: 16 Rounds Ford No 2
Census Place: Oldbury, Worcester, England
Source: FHL Film 1341680 PRO Ref RG11 Piece 2838 Folio 83 Page 13
Marr Age Sex Birthplace
Enoch SPENCER M 28 M West Bromwich, Stafford, England
Rel: Head
Occ: Coal Wheeler (Miner)
Fanny SPENCER M 25 F Quarry Bank, Stafford, England
Rel: Wife
Fanny SPENCER 3 F Oldbury, Worcester, England
Rel: Daur
Joseph SPENCER 1 M Oldbury, Worcester, England
Rel: Son
There are 48 Joseph Spencers in this region but the only Joseph Spencer born in W.Brom & living in the region is this one, but the wife's name is different:
Dwelling: 12 Sandwell Rd
Census Place: West Bromwich, Stafford, England
Source: FHL Film 1341683 PRO Ref RG11 Piece 2850 Folio 44 Page 32
Marr Age Sex Birthplace
Joseph SPENCER M 42 M West Bromwich, Stafford, England
Rel: Head
Occ: Horse Agent & Vaccination Offc
Eliza SPENCER M 49 F Sedgley, Stafford, England
Rel: Wife
Eliza J. SPENCER U 21 F West Bromwich, Stafford, England
Rel: Daur
Occ: Domestic
Lydia SPENCER U 19 F West Bromwich, Stafford, England
Rel: Daur
Occ: Dress Maker
Emma SPENCER U 16 F Sheffield, York, England
Rel: Daur
Occ: Dress Maker Assistant
George H. SPENCER U 14 M Sheffield, York, England
Rel: Son
Occ: Scholar
I also Found this information within the Free BMD Website:
Births Mar 1852
Spencer Joseph W Bromwich 6b 582
Births Jun 1853
Spencer Enoch W Bromwich 6b 532
Births Dec 1881
SPENCER Enoch W. Bromwich 6b 738
Is this any help to you at all?