Hello here are the following entries for the names John and Jemima Bird  
 
 No Match to Jemima in 1841 (do you have a maiden name for her?) 
There is another John Bird besides this one, but living in Swansea S.Wales c.1811 (not born in Glamorgan) no other matches:
Glamorgan Family History Society 1841 Census CD
District Name: Cardiff             Enumeration District:  4 Cardiff                
Piece: 1425/03      Folio: 028     Schedule: 192A
Address: WHITMORE LANE          
C 21 028 192A WHITMORE LANE          BIRD JOHN                  27   (c1814) 
                                     LABOURER                        NO  
C 21                                 JONES JOHN                 30   (c1811) 
                                     NAILER                          NO  
C 21                                 KELLY MARY                 40   (c1801) 
                                     -                               IRL 
______________________________________
2455 1851 CENSUS OF CARDIFF (NORTH-EAST). REF.H.O.107/2455, E.D.4B.                                                                  
Piece: 2455      Folio: 101     Schedule: 030
Address: SPRING GARDEN VILLA    
BIRD JOHN                  HD M  35   ATTORNEY AT LAW           
                                      GLA COWBRIDGE                    (c1816)
BIRD JEMIMA                WI M  35   -                         
                                      MDX LONDON                       (c1816)
BIRD JOHN J.S.             SO -  10   -                         
                                      GLA CARDIFF                      (c1841)
BIRD MIRA ANN              DA -   8   -                         
                                      MDX CHISWICK                     (c1843)
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Glamorgan Family History Society 1861 Census CD
4032 1861 CENSUS OF CARDIFF ST.JOHN. REF.R.G.9/4032, E.D.7.                                                                          
Piece: 4032      Folio: 092     Schedule: 275
Address: SPRING GARDEN HOUSE    
BIRD JOHN                  HD W  44   ATTORNEY AT LAW           
                                      GLA COWBRIDGE                    (c1817)
STRUTT MARY A.             ML W  76   FUND & SHAREHOLDER        
                                      HRT HERTFORD                     (c1785)
STRUTT KEZIA               CL U  41   FUNDHOLDER               [
                                      MDX]ISLEWORTH                    (c1820)
BIRD MIRA A.               DA U  18   -                         
                                      MDX CHISWICK                     (c1843)
SILCOX JOHN W.             VR U  25   WESLEYAN MINISTER         
                                      GLS BRISTOL                      (c1836)
PARTRIDGE MARY             SV W  28   DOMESTIC SERVANT          
                                      PEM CASTLE MARTIN                (c1833)
________________________________________
John is listed as a Widow in 1861, so Jemima died between 1851 - 1861. I found a death entry for a Jemima Bird on the Free BMD site:
Deaths Jun Quarter 1858
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Bird  Jemima     Cardiff  11a 160 
____________________________________________
I also found a marriage entry for a John Bird a few years after this death entry, could your John have re-married 9 yrs later? 
Marriages Sep 1867
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BIRD  John     Cardiff  11a 371 
There are 2 women in the same transcription (it could be the wife of this individual hasn't been transcribed on the Free BMD's as of yet):  
Fowler  Elizabeth     Cardiff  11a 371   
Williams  Sarah     Cardiff  11a 371   
If you notice they have a Wesleyan Minister visiting them in 1861 at the time of the census, They were in all probability Wesleyan Methodists.
I hope this helps you
Kind Regards Jen  

NB: "This lookup was made using material published by Glamorgan Family History
Society, details of membership and their research service with a full list
of published indexes and transcriptions for sale can be seen on
http://www.glamfhs.org/ "