I think you can probably eliminate the William Wilkins who married Jane Dadd Rogers. Other than the 1871 census the only other references to his occupation I could find were as a shipwright.
GRO BMD references:1838 Q1 Birth - Jane Dadd Rogers (mmn: Richardson); Poplar; 02; 268
1859 Q2 Marriage - William Wilkins & Jane Dad Rogers; Hackney; 01b; 445
1891 Q1 Death - William Wilkins; age 52 (born c1839); Hackney; 01b; 402
1920 Q2 Death - Jane Dadd Wilkins; age 82 (born c1838); West Ham; 04a; 228
1864 Q4 Birth - Harriet Wilkins (mmn: Rogers); Poplar; 01c; 611
1869 Q3 Birth - Constance Ethel Wilkins (mmn: Rogers); Mile End Old Town; 01c; 534
1871 Q3 Birth - William James Wilkins (mmn: Rogers); Poplar; 01c; 582
1880 Q3 Birth - Gertrude Elizabeth Wilkins (mmn: Rogers); Poplar; 01c; 613
1882 Q1 Birth - Annie Louise Wilkins (mmn: Rogers); Poplar; 01c; 776
1898 Q3 Marriage - William Merrell & Constance Ethel Wilkins; Hackney; 01b; 1150
1899 Q3 Birth - Harold William Gannaway (mmn: Wilkins); Hackney; 01b; 527
1901 Q1 Birth - Ethel Winifred Gannaway (mmn: Wilkins); Hackney; 01b; 482
Marriages:married: 25 Jun 1859; St John, Hackney; lic
groom: William Wilkins; fa; bach; Shipwright; Poplar
bride: Jane Dad Rogers; fa; spin; -; Church Street
grooms father: Charles Weston Wilkins, shipwright
brides father: Martin Dad Rogers, shipwright
witnesses: William Henson, Emma? Elizabeth Wilkins
married: 30 Jul 1898; St Paul, Homerton, Hackney; banns
groom: William Merrell Gannaway; 29; bach; wood machinest?; 22 Durrington Rd
bride: Constance Ethel Wilkins; 29; spin; -; 22 Durrington Rd
grooms father: William Henry Gannawy, laundry man
brides father: William Wilkins, shipwright
witnesses: Harriott Wilkins, William James Wilkins
Censuses:1861 Poplar
William (shipwright) and Jane
Surname difficult to read, could be Wilkins.
Ref RG09; Piece 300; Folio 82; Page 9;
https://www.familysearch.org/ark:/61903/1:1:Q2ML-G1JS1871 Poplar
William (dealer in ?), Jane and 2 daughters.
Ref RG10; Piece 573; Folio 78; Page 40;
https://www.familysearch.org/ark:/61903/1:1:VB6W-F1M1881 Poplar
Jane (married), 2 daughters and 1 son.
Ref RG11; Piece 502; Folio 80; Page 26;
https://www.familysearch.org/ark:/61903/1:1:Q27S-Y8ZL1891 Hackney
Jane (widow), 3 daughters and 1 son
Ref RG12; Piece 207; Folio 73; Page 10;
https://www.familysearch.org/ark:/61903/1:1:QWFR-DPZ1901 Hackney
Jane (married?), 2 daughters, 1 son-in-law (William Grannaway/Gannaway) and 1 grandson.
Ref RG13; Piece 221; Folio 49; Page 20;
https://www.familysearch.org/ark:/61903/1:1:X9ZW-S551911 Hackney
Jane (widow)
https://www.familysearch.org/ark:/61903/1:1:XWGC-4GFNewspaper items:Hackney and Kingsland Gazette
13 Feb 1891
Shocking Suicide In Hackney Infirmary
Yesterday afternoon Dr Macdonald, M.P., held an inquest at the Hackney Union Workhouse, Homerton, on the body of William Wilkins, a shipwright, who cut his throat in the Hackney Union Infirmary on Sunday last.
William James Wilkins, 4, Gainsborough square, South Hackney, labourer, said the deceased was his father, and had been blind for 12 or 14 years. He lost his sight gradually. Witness went to the Union on Sunday to see the deceased. He stayed about an hour. Deceased seemed rather depressed, but not more so than usual. He asked witness to lend him his pen-knife. Witness did so, having no suspicion that he wanted to do himself an injury. When he got outside he thought he had done wrong, and on the advice of his friend he went back and told the porter at the gate. A short time afterwards he heard that the deceased had cut his throat. Deceased had never threatened to commit suicide. He had been a little light headed at times after his eyes had been attended to.
Rose Freeman, a nurse in the Infirmary, said the deceased was under her charge. She did not know he had the knife until she was told so by the porter on Sunday afternoon. As soon as she heard of it she went to the deceased, who at first denied having it, but subsequently admitted that he had. Witness asked him to give it up, remarking that he could have no use for it as his food was cut up for him. He refused to let her have it, and as she turned to speak to the porter he suddenly produced the knife, and exclaimed, 'Can't I do anything with it,' and before she could reach him he had cut his throat with it. He died almost immediately.
William Shopland said the son of the deceased informed him on Sunday afternoon that he had lent his father a knife, and witness went and told the nurse. As she was endeavouring to get it from him he cut his throat with it before he could be stopped.
Dr Gordon said he had had the deceased under observation for some time, but could not detect any sign of insanity. He was merely suffering from mental depression. On the afternoon in question witness was informed that he had cut his throat. Witness at once went to him but could be of no service, expiring almost immediately. All the chief arteries and the windpipe were severed.
The jury returned a verdict of 'Suicide whilst of unsound mind.'