His baptism:

First name(s) James Boardman
Last name Cartwright
Gender Male
Birth year 1797
Baptism year 1797
Baptism date 15 Aug 1797
Residence Manchester, Lancashire, England
Place Manchester
County Lancashire
Country England
Father's first name(s) Robert
Father's last name Cartwright
Mother's first name(s) Mary
Record set England Births & Baptisms 1538-1975
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From Venns’s Alumni Cantabriensis:

CARTWRIGHT, JAMES BOARDMAN. Adm. sizar at Queens’ June 17 1816. Matric. Michs. 1816; B.A. 1821; M.A. 1824. Ord. deacon, 1820; priest (Lincoln) 1821. Minister of the Jewish Episcopal Chruch, Bethnal Green, c. 1831-61. Chaplain and Principal of the Hebrew College, in 1853-9-. Died Feb 8 1861, aged 63, in Bethnal Green. (G. Mag., 1862, 1. 347.)
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His death index from FreeBMD as FindMypast does not have this entry:

Surname First name(s) Age District Vol Page
Deaths Mar 1861   (>99%)
CARTWRIGHT James Breadman Bethnal Gn 1c 231
Cartwright James Broadman Bethnal Gn 1c 231
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His 1851 census entry, with the same two children as in 1841:

First name(s) Last name Relationship Marital status  Gender  Age  Birth year  Occupation  Birth place

James Bourdmen Cartwright Head Married Male 54 1797  Cleyyman Ch Of Engld Minister Of Epi?cossal ? Chapel Manchester, Lancashire
Sophia Cartwright Wife Married Female  54  1797  -  Wellington, Shropshire
Elizabeth Cartwright Daughter Unmarried Female  24  1827  -  London, Middlesex
William Lyster Cartwright Son Unmarried Male 14  1837  Scholar  Bethnal Green, Middlesex
Laura Andrew Cartwright Visitor Unmarried Female  34  1817  -  Northamptonshire
Sophia Cousins Servant Unmarried Female 28 1823  General Servant  Hackney, Middlesex
Sarah Hatch Clayton Servant Unmarried Female 26  1825  General Servant  Southwark, Surrey

Address:
Street Palestine Place
Parish Bethnal Green
City Tower Hamlets
County Middlesex

Census:
Archive reference HO107
Piece number 1540
Folio 601
Page 30
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From the CCED, Clergy of the Church of England Database, his appointments were:

31 Dec 1820: ordained deacon
13 Jan 1822: ordained priest
31 Dec 1820: Stipendiary Curate at Butterwick, Lincs
10 Jun 1822: Stipendiary Curate of Fleet, Lincs
30 Aug 1824: Perpetual Curate of Bierley chapel, Bradford, to 20 Oct 1826
31 Mar 1832: Minister at the St Matthew Bethnal Green Chapel of the London
               Society for promoting Christianity amongst the jews.
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So far no information has been fround to show whether he went to Australia with the rest of the family.  He would have been 12 at the time of departure so might well have been at a school and he could have stayed with a relation in England.
On the other hand he did not get into Cambridge until he was 19 which was slightly later than customary and might have been accounted from by the delay of a boat trip back from Ausgtralia.
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His probate index:

CARTWRIGHT The Reverend James Boardman
Effects under £450

13 March. The Will of the Reverend James Boardman Cartwright late of 1 Palestine place Bethnal Green in the County of Middlesex Clerk deceased who died 8 February at Palestine place aforesaid was proved at the Principal Registry by the oath of Sophia Cartwright of 1 Palestine place aforesaid Widow the Relect the sole Executrix.
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Their marriage:

First name(s) James Boardman
Last name Cartwright
Marriage year 1825
Marriage date 21 Oct 1825
Marriage place Wellington, All Saints
Denomination Anglican
Residence Bradford, Yorkshire
Spouse's first name(s) Sophia
Spouse's last name Cartwright
Spouse's residence Wellington, All Saints
County Shropshire
Register date range 1813-1827
Archive reference P291/A/3/5
Page 244
Register type Marriage
Record set Shropshire marriages
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