His birth, from Modern IGI:

name: Thomas Oliver Watts Coster
gender: Male
baptism/christening date: 26 Nov 1816
baptism/christening place: ST GEORGE THE MARTYR,SOUTHWARK,LONDON,ENGLAND
birth date: 21 Apr 1816
father's name: Thomas Coster
mother's name: Maria
indexing project (batch) number: C02244-1
system origin: England-ODM
source film number: 307691
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His death:

Surname  First name(s)  Age  District  Vol  Page 
Deaths Mar 1886   (>99%)
Coster Thomas Oliver W 69 Marylebone 1a 486
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Entries in the London Gazette:

Issue 19529, published on 4 Aug 1837, p. 3 of 32: War Office on 4th Aug 1837: 4th Regiment of Dragoon Guards, Thomas Oliver Watts Coster, Gent, to be Cornet, by purchase, vice Prendergast, whose appointment had not taken place.

Issue 20587, published on 27 Mar 1846, p. 17 of 48: War Office on 27th March 1846: 4th Dragoon Guards, Lieutenant Thomas Oliver Wells Coster to be Captain, by purchase, vice Rochfort, who retirees.
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From the 1871 census:

Surname  Forename  Age  Relation  Occupation  Birth Place

Coster Thomas O W 54 Head Late Capt in 4th Dragoons  London
Coster Mary E P 41 Wife India
Coster Marian 16 Daur London
Coster Arthur A 15 Son London
Coster Churchill W  13 Son London
Coster Ellen 10 Daur London
Spanton Emily   22 Servant Loughborough, Leicester
Maynard Alfred 19 Servant Tunbridge Wells, Sussex  
Delve Mary Ann 24 Servant Bow, Devon
Campfield Louisa 23 Servant Brighton, Sussex
Royall Matilda 20 Servant London
Hort Sarah 26 Servant Shillingstone, Dorset
Lichfield Sarah 26 Servant London
Moseley Elizabeth 17 Servant London
Finch Lucia 20 Servant Caple, Suffolk

Address: 137 Harley Street, Portland Place, Marylebone, London

Ref: RG10/0157/~F20
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I wonder if this was also a child of theirs:

Surname   First name(s)      District   Vol  Page
Births Dec 1855   (>99%)
Coster Mary Ellen    Marylebone 1a 345

No, not possible as Arthur Augustus was born 5 months later!
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He was also in the 1881 census as a widower and, with his two sons, of ‘no occupation’ with a 26-year old niece by his wife.
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His probate index:

“COSTER Thomas Oliver Watts Esq
“Personal Estate £67,839 1s 10d
“18 April [1886].  The Will with a codicil of Thomas Oliver Watts Coster late of 137 Harley-Street in the parish of St. Marylebone in the County of Middlesex Esquire who died 27 March 1886 at 137 Harley-street was proved at the Principal Registry by Arthur Augustine Coster and Churchill William Coster both of 137 Harley-street Esquires the Sons the Executors”
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Their marriage:

Surname   First name(s)      District   Vol  Page
Marriages Mar 1852   (>99%)
Coster Thomas Oliver Watts Marylebone 1a  518
Davies Mary Ellen Parry    Marylebone 1a 518