His birth index:

First name(s)         William Alexander
Last name             Carroll
Birth quarter         2
Birth year            1863
Mother's maiden name  Park
District              Wandsworth
County                London
Country               England
Volume                1D
Page                  487
Record set            England & Wales Births 1837-2006
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In 1901 he was married and with a new-born daughter and had both his mother and his brother with them at (St George's Vicarage), Bickley Park Road, Bromley, Kent:

First name(s)  Last name  Relationship  Marital status  Sex  Age  Birth year  Occupation  Birth place
William A  Carroll   Head     Married  Male    37  1864  Clergyman (Church Of England)   Putney, Surrey
Nora J B   Carroll   Wife     Married  Female  25  1876  -                               Scotland
Delia M C  Carroll   Daughter  -       Female   0  1901  -                               Bickley, Kent
Lucy       Carroll   Mother   Widow    Female  70  1831  -                               London
Hugh D     Carroll   Brother  Single   Male    36  1865  Manufacturer (Electric Compos)  Putney, Surrey
Bertha M   Sadler    Servant  Single   Female  35  1866  Nurse Domestic                  Sonning, Berkshire
Mary A     Kilby     Servant  Single   Female  41  1860  Lady's Maid Domestic            Marsh Gibbon, Bucks
Edith      Kemp      Servant  Single   Female  21  1880  Lady's Maid Domestic            Hampshire, England
Lucy S     Arnold    Servant  Single   Female  50  1851  Cook Domestic                   Caerwent, Monmouths
Ellen M    Cannings  Servant  Single   Female  32  1869  Parlormaid Domestic             Cudham, Kent
Edith      Stacey    Servant  Single   Female  23  1878  Housemaid Domestic              Farnham, Surrey
Sadia      Wellings  Servant  Single   Female  15  1886  Housemaid Domestic              Northwich, Cheshire

Ref:
Archive reference  RG13
Piece number       683
Folio              112
Page               2
Combined:          RG13, piece 683, folio 112, p. 2
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The following births were found on the new GRO index in the Bromley district:

Name:                           Mother's Maiden Surname:
CARROLL, -                      HAMILTON  
GRO Reference: 1900  J Quarter in BROMLEY  Volume 02A  Page 473

CARROLL, URSULA MARY JEAN       HAMILTON  
GRO Reference: 1908  S Quarter in BROMLEY  Volume 02A  Page 571

CARROLL, MARY  STELLA HENRIETTA  HAMILTON  
GRO Reference: 1910  M Quarter in BROMLEY  Volume 02A  Page 532

CARROLL, MARY  GRISELDA LUCY     HAMILTON
GRO Reference: 1914  M Quarter in BROMLEY  Volume 02A  Page 1058

The 1911 census confirms the first three of these children, the fourth being born three years later.
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His death index:

First name(s)  William A
Last name      Carroll
Gender         Male
Birth year     1863
Age            72
Death quarter  3
Death year     1935
District       Oxford
County         Oxfordshire
Volume         3A
Page           1144
Country        England
Record set     England & Wales Deaths 1837-2007
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His probate index:

1935
CARROLL the reverend canon William Alexander of the Warneford Mantal Hospital Oxford clerk died 16 August 1935  Probate Oxford 20 November to Claud Archibald Mackenzie Bruce Hamilton of no occupation and John Colin Howard Lindsay retired captain R.N.  Effects £672 2s. 5d.
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Their marriage on Scotlands People:

Surname Forename         Spouse Surname  Spouse Forename    Year  Ref    Entry Number  RD Name
HAMILTON  NORA JANE BRUCE  CARROLL         WILLIAM ALEXANDER  1899  469/2  2             Airth

They were second cousins with common ancestors of John and Jean, née Mayne, Brown.