No death record on FreeBMD.  This has to be because he was still alive, even though Burke’s 1958 for Ireland said he died in 1877.  He appears as the Head of Household in the 1881 census with his wife, children and five household staff:

Name         Relation  Marital Status  Gender  Age  Birthplace  Occupation

Henry James COLDHAM Head M Male 41 Anmer, Norfolk   Magistrate Farmer 500 Acres Employing 30 Men 11 Boys
Agatha Geraldine COLDHAM Wife M Female 30 Wold, Northampton
James N.N. COLDHAM    Son    Male 7 Anmer, Norfolk
Rhoda Geraldine COLDHAM Dau    Female 5 Anmer, Norfolk
Frances Elizabeth COLDHAM Dau    Female 3   Anmer, Norfolk
Agatha Maria COLDHAM    Dau    Female 1 Anmer, Norfolk
Rachel WILSON    Servant U Female 28 Easton, Norfolk  Cook Domestic Serv
Eliza THOMPSON    Servant U Female 26 Ipswich, Suffolk  Nurse Domestic Serv
Mary Ann JARVIES Servant U Female 30 Durham, Durham  Parlour Maid Domestic Serv
Harriett Elizabeth TERRING Servant U Female   17   Stanhoe, Norfolk  Nursemaid Domestic Serv
Mary Ann EBBAGE       Servant U Female 17 Foulsham, Norfolk  Kitchenmaid Domestic Serv
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However FreeBMD has a Henry James who d. in Docking, Norfolk dist in 1887 so I’ve taken that to be him.  (100% positive, too! Confirmed by the probate entry.)

First name(s) HENRY JAMES
Last name COLDHAM
Gender Male
Birth year 1840
Age 47
Death quarter 3
Death year 1887
District Docking
County Norfolk
Volume 4B
Page 193
Country England
Record set England & Wales Deaths 1837-2007
___________________

His burial:

First name(s) Henry James
Last name Coldham
Sex Male
Age 47
Birth year 1840
Burial year 1887
Burial date 16 Jul 1887
Parish Anmer
Diocese Norwich
County Norfolk
Country England
Archive Norfolk Record Office
Archive reference PD 498/24
Page 23
Record set Norfolk Burials
_________________

His probate record:

COLDHAM Henry James Esq
2 November
Personal Estate £12350 6s 5d
The will with a codicil of Henry James Coldham late of Anmer Hall in Anmer in the County of Norfolk Esquire who died 13 July 1887 at Anmer was proved at Norwich by George Francis Addison Cresswell of King’s Lynn in the said County Banker and George John Coldham of 5 New Inn inthe County of Middlesex Gentleman the Executors.
__________________

Arms: Azure a mullet Argent pierced Gules.

Crest: A griffin’s head couped, pierced through by an arrow.
______________

In the 1634 visitation of susses, there were some Coldhanms with the same arms but with a crescent or in dexter chief for difference.

But no Coldham entry in the 1664 visitation of Norfolk.
______________

Burke’s Landed Gentry of 1871, vol I, p. 259 and of 1847/8, vol 3, p. 67

Their marriage:

Surname First name(s) District Vol Page
Marriages Sep 1870   (>99%)
Hamilton Agatha Geraldine Docking 4b 561

Surname   Given Name   District   Volume   Page
Marriages Sep 1870
Coldham Henry James Dorking 2a 561   <--- vol=2b is wrongly transcribed, is 4b