This entry has been found on the new GRO birth indices which could, just, be his:

Name                 Mother's Maiden Surname
KENRICK, MALE     -  
GRO Reference: 1837  S Quarter in WREXHAM  Volume 27  Page 169
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No baptism can be found in Denbighshire baptisms.
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At marriage he clearly wrote his second name as 'Wynn'.
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Graeme Wall found these children and shows them on his website:

Lydia Marian, 1862-1943
Hubert Wynn, 1863-1939
Annie Sara, 1865-
John Painter, 1865-1946
Mary Katharine, 1867-1952
William Hamilton, 1868-1927
Minnie, Myfanwy, 1870-1957
Muriel Yerbury, 1878-1969

All the Birth registrations have been found and with the mother's maiden name of James with the exception of Hubert Wynn's which is missing from the new GRO indices.  The GRO have been told of this lacuni.

Only the 1881 census has been found with the father, William, in residence.
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Eventually, after finding the 1881 census, his family was found, without him, in the 1771 census at Hall Street, Bache, Corwen, Merionethshire, Wales:

First name(s) Last name Relationship Marital status  Sex  Age  Birth year  Occupation Birth place
Mary Ann Kendrick Head - Female 34 1837 - Denbighshire, Wales
Herbert Wynne Kendrick Son - Male 7 1864 - Denbighshire, Wales
Anne Sara Kendrick Daughter - Female 6 1865 -  Denbighshire, Wales
John Painter Kendrick Son - Male 5 1866 - Denbighshire, Wales
Mary Katherine Kendrick Daughter - Female 4 1867  -  Staffordshire, England
William Hamilton Kendrick Son - Male 2 1869 - Denbighshire, Wales
Mariner Myfanwy Kendrick Daughter - Female 0 1871  -  Denbighshire, Wales
Petronella James Grandmother - Female 91 1780  -  Middlesex, England
Katherine James Sister - Female 31 1840 -  Denbighshire, Wales
Catherine Ellis Servant - Female 27 1844 -  Denbighshire, Wales
Sarah Gundy Servant - Female 17 1854 - Denbighshire, Wales

Ref:
Archive reference  RG10
Piece number       5683
Folio              48
Page               14
Combined:          RG10, piece 5683, folio 48, p. 14

Note that Lydia Marian was not with them as she was away at a boarding school in Wem, Shropshire.

Additionall this is the first public record to be found for Hubert, though here written as Herbert, Wynn.
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However (to the thoughts they they could not be found) this is them in the 1881 census at the Oaklands, Rugely, Lichfield, Staffs:

First name(s)  Last name  Relationship  Marital status  Sex  Age  Birth year  Occupation  Birth place
William Kendrick Head Married Male 43 1838 Mining Engineer  Ruabon, Denbighshire, Wales
Mary Ann Kendrick Wife Married Female 44 1837 - Wrexham, Denbighshire, Wales
Marian Kendrick Daughter Single Female 19 1862 -  Denbighshire, Wales
Mary Catherine Kendrick Daughter Single Female 14  1867  Scholar  Walsall, Staffordshire
Minnie Kendrick Daughter Single Female 10 1871 Scholar  Llangollen, Denbighshire, Wales
Muriel Kendrick Daughter Single Female 7 1874 -  St Fagans, Glamorganshire, Wales
Leah Jones Servant Single Female 18 1863 Servant  Chepstow, Monmouthshire, Wales
Lucy Cox Servant Single Female 14 1867 Servant Rugeley, Staffordshire

Ref:
Archive reference  RG11
Piece number       2779
Folio              82
Page               9
Combined:          RG11, piece 2779, folio 82, p. 9
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After death he was descrbed in family notices in newspapers as Government Commissioner of Mines for Demerara, British Guina.  See <http://sites.rootsweb.com/~nyggbs/Transcriptions/LisaB/MarriagesTrans2015.pdf>;.
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This notice of his death was found on a British Guiana website in 2019.  It is a report from The Times of 6 Oct 1892:

KENRICK - A very painful sensation was created, both in general and official circles in British Guiana, on September 17, by the tidings which were brought to Georgetown from the north-west district to the effect that Mr. W. W. KENRICK, Commissioner of Mines for the colony, had died from bilious remittent fever at the Government agent's house at Morawhanna, where he went at the end of August for the purpose of making an official inspection and reporting upon the goldfields of the north-western portion of the colony. Mr. Kenrick was appointed by Lord Knutsford to be Gold Commissioner of the colony - his office subsequently came to be known by the title of the Commissionership of Mines - and came to British Guiana in January,
1891. He was closely concerned with the framing of the new mining regulations now in force in the colony, having had large experience of the industry in South Africa and California. He made several expeditions into the interior with the object of reporting upon the wealth of the colony, and on the occasion of his last journey he and his party had got as far as the Amacurvo r... [rest missing]

As of 2 Oct 2019, I could not get a download of this article from The Times' archives.
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Their marriage index:

First name(s)    WILLIAM WYNN
Last name        KENRICK
Marriage quarter 2
Marriage year    1861
MarriageFinder   WILLIAM WYNN KENRICK married one of these people
                 Sarah Jones, Mary Anne James
District         Wrexham
County           Denbighshire
Country          Wales
Volume           11B
Page number      541
Record set       England & Wales Marriages 1837-2005
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The parish register abstract is:

First name(s)                     William Wynn
Last name                         Kenrick
Age                               Full
Event year                        1861
Marriage year                     1861
Marriage date                     20 Jun 1861
Marriage place                    Wrexham
Father's first name(s)            William
Father's last name                Kenrick
Spouse's first name(s)            Mary Anne
Spouse's last name                James
Spouse's age                      Full
Spouse's father's last name       James
Spouse's father's first name(s)   John
County                            Denbighshire
Country                           Wales
Event                             Marriage
Page                              11
Archive type                      Marriages
Archive year range                1861-1866
Record set                        Denbighshire Marriages And Banns

from the scan of the register page is added:

His job:        Smelter
Her job:        -
His residence:  Ruabon
Her residence:  Wrexham
Marriage by:    Licence
His pa's job:   Iron and coal master
Her pa's job:   Gentleman
Celebrant:      George Cunliffe vicar
Witnesses:      Thos G Crane(?), Jno James, Aarah S Kenrick, John Kenrick
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