Curiously FHS Film 1042307 in LDS Vital records, edn 2, records a birth on 3rd Jul 1711 of William Cook, son of Thomas and Mary Cook and christened on 8 Jul 1711 at St James' Westminster.

And FamilySearch has this transcription, but no original:

Name:                          William Cooke
Gender:                        Male
Christening Date:              15 Jul 1711
Christening Place:             SAINT PETER PAULS WHARF,LONDON,LONDON,ENGLAND
Birth Date:                    02 Jul 1711
Father's Name:                 William Cooke
Mother's Name:                 Ann
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See Thomas Harwood’s “Alumni Etonienses”, pp. 50 and 320:

“William Cooke, Mar 25 1772, Geo III. 12.  The present learned Provost of King’s College was admitted a Scholar in 1730, was A.B. 1734, A.M. 1738, S.T.P. 1766.  On the 20th May 1743, he was elected head Master of Eton School, which office he did not hold two years.  He was Chaplain to the Earl of Halifax, and in 1743, preferred by Mrs Edwin to the rectory of Denham, in Bucks.  In 1745, he was presented by Eton College to the living of Stourminster Marshall, in Dorsetshire. In 1750, he published a Visitation Sermon preached at Beaconsfield.  In 1747 was Fellow of Eton College and in 1780, published Concio ad Clerum.  He is now Dean of Ely, Rector of Denham, in Bucks, and of Stoke Newington in Middlesex.”
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I wonder if this was his burial:

name: William Cooke
gender: Male
burial date: 28 Oct 1797
burial place: Denham, Buckingham, England
indexing project (batch) number: I01113-3
system origin: England-EASy
source film number: 1042778
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Cooke, William 1711-1797

Name: Cooke, William
Dates: 1711-1797
Active Date: 1751
Gender: Male

Field of Interest: Education, Scholarship and Languages
Occupation: Provost of King's College, Cambridge
Place of
    Birth: St. James's, Westminster
    Education: Harrow,   Eton,   King's College, Cambridge
    Death: Bath
Spouse: Catherine, daughter of Richard Sleech
Sources: Nichols's Lit. Anecd. ix. 629, 630; Harwood's Alumni Etonenses, p...
Contributor: L. S. [Leslie Stephen]

Co-subject: Cooke, William
Dates: c.1749-1824
Active Date: 1789
Gender: Male
Field of Interest: Scholarship and Languages
Occupation: Greek professor

Article
Cooke, William 1711-1797, provost of King's College, Cambridge, was born in St. James's, Westminster, 15 Oct. 1711. He was sent to Harrow in 1718, and placed upon the foundation at Eton in 1721. In 1731 he became a scholar, and in 1734 a fellow, of King's College, Cambridge. He graduated B.A. in 1735, and soon afterwards became an assistant-master at Eton. In May 1743 he was unanimously elected head-master, but found his health too weak for the place, and in 1745 took the college living of Sturminster-Marshall, Dorsetshire. In 1748 he was elected fellow of Eton College, and resigned Sturminster on being presented to the rectory of Denham, Buckinghamshire; he was also bursar of Eton. In 1765 he proceeded D.D., and was appointed chaplain to the Earl of Halifax. In 1768 he accepted the rectory of Stoke Newington. On 25 March 1772 he was unanimously elected provost of King's College, Cambridge. He was vice-chancellor of the university in 1773. In April 1780 he received a prebend in Ely, and on 9 Aug. was appointed to the deanery. He died at Bath 20 Oct. 1797.
He married Catherine, daughter of Richard Sleech, canon of Windsor, in January 1746, and had by her twelve children. His second daughter, Catherine, married Bishop Samuel Hallifax [q.v.], whose epitaph was written by Cooke. Cooke published a few sermons, and in 1732 a small (anonymous) collection of poems called `Mus‘ Juveniles,' including a Greek tragedy upon Solomon, called ÷ñ<”p Ïx‰‚pûñ}. In one of the sermons (1750) upon the meaning of the expression in the second Epistle of St. Peter, `a more sure word of prophecy,' he defends Sherlock against Conyers Middleton, and produced a little controversy. He composed an epitaph for himself in a south vestry of King's College Chapel, attributing whatever he had done to the munificence of Henry VI. One of his sons, Edward Cooke [q.v.], became secretary at war in Ireland. Another son, William Cooke, was fellow of King's College, Cambridge, professor of Greek at Cambridge from 1780 to 1792, and rector of Hempstead-with-Lessingham, Norfolk, from 1785 till his death, 3 May 1824. He published an edition of Aristotle's `Poetics' in 1785, to which was appended the first translation of Gray's `Elegy' into Greek verse, a performance which had many imitators at the time (Nichols, Lit. Anecd. ix. 154-5). Mathias praises Cooke's translation as equal to Bion or Moschus, and calls the author an `extraordinary genius' (Pursuits of Literature, Dial. iii.); but De Quincey in `Coleridge and Opium Eating' declares that `scores of modern schoolboys' could do as well. In 1789 he also published `A Dissertation on the Revelation of St. John,' comparing the Apocalypse to the `fdipus Tyrannus' of Sophocles and to Homer. He verified the old saying as to the result of such studies by afterwards becoming deranged (Gent. Mag. for 1798, p. 774, and 1824, ii. 183).

Sources
Nichols's Lit. Anecd. ix. 629, 630; Harwood's Alumni Etonenses, p. 50; Le Neve's Fasti, i. 349, 357; Gent. Mag. 1797, ii. 901, 953.

Contributor: L. S.

published 1887

Their marriage on FamilySearch:

groom's name: William Cooke
bride's name: Catharine Sleech
marriage date: 28 Jan 1745
marriage place: Saint Martin In The Fields,Westminster,London,England
indexing project (batch) number: M00145-7
system origin: England-ODM
source film number: 561155
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