1851 census
Name: Joseph Braithwaite
Age: 3
Estimated Birth Year: abt 1848
Relation: Son (Child)
Father's name: James Braithwaite
Mother's name: Sarah Braithwaite
Gender: Male
Where born: Cliburn, Westmorland, England

James Braithwaite 22 occ miller /grocer b Melmerby, Cumberland, England
Sarah Braithwaite 20 b Cliburn, Westmorland
Joseph Braithwaite 3 b Cliburn, Westmorland
John Braithwaite 7 Mo b Cliburn, Westmorland

England Births and Christenings
Name: Joseph Braithwaite
Gender: Male
Christening Date: 20 Feb 1848
Christening Place: Cliburn, Westmoreland, England
Father's Name: James Braithwaite
Mother's Name: Sarah
Indexing Project (Batch) Number: C02840-8 , System Origin: England-EASy , GS Film number: 1471666 , Reference ID: item 10 p 30

This looks like Joseph's parents marriage

Marriage
Name: James Braithwaite
Birth Date: 1828
Age: 20
Spouse's Name: Sarah Mcdonell
Spouse's Birth Date: 1831
Spouse's Age: 17
Event Date: 1848
Event Place: Cliburn, Westmoreland, England
Father's Name: Thomas Braithwaite
Spouse's Father's Name: Donald Mcdonell

Indexing Project (Batch) Number: M00318-3 , System Origin: England-VR , GS Film number: 1471666 , Reference ID: 2:3CVP7FB

England Births and Christenings
Name: John Braithwaite
Gender: Male
Christening Date: 06 Oct 1850
Christening Place: CLIBURN,WESTMORLAND,ENGLAND
Father's Name: James Braithwaite
Mother's Name: Sarah
Indexing Project (Batch) Number: C00318-2 , System Origin: England-ODM , GS Film number: 97362

England Births and Christenings
Name: Joseph Braithwaite
Gender: Male
Christening Date: 20 Feb 1848
Christening Place: Cliburn, Westmoreland, England
Father's Name: James Braithwaite
Mother's Name: Sarah
Indexing Project (Batch) Number: C02840-8 , System Origin: England-EASy , GS Film number: 1471666 , Reference ID: item 10 p 30

England Births and Christenings
Name: James Braithwaite
Gender: Male
Christening Date: 20 Jul 1828
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

Marriage
Name: Thomas Braithwaite
Spouse's Name: Hannah Barrow
Event Date: 03 Dec 1819
Event Place: Melmerby,Cumberland,England
Indexing Project (Batch) Number: M06092-1 , System Origin: England-ODM , GS Film number: 90646

other children:

England Births and Christenings
Name: John Braithwaite
Gender: Male
Christening Date: 16 Oct 1831
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Watson Braithwaite
Gender: Male
Christening Date: 11 Apr 1845
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Birth Date: 11 Aug 1844
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Hannah Braithwaite
Gender: Female
Christening Date: 04 Feb 1827
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Thomas Braithwaite
Gender: Male
Christening Date: 21 Jun 1835
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: John Braithwaite
Gender: Male
Christening Date: 06 Apr 1820
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Death Date: 23 Aug 1829
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Elisabeth Braithwaite
Gender: Female
Christening Date: 29 Jan 1837
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Death Date: 05 Jun 1837
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Jane Braithwaite
Gender: Female
Christening Date: 11 Aug 1833
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Elisabeth Braithwaite
Gender: Female
Christening Date: 26 May 1830
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Death Date: 07 Jun 1830
Father's Name: Thomas Braithwaite
Mother's Name: Hannah
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

1841 Census
Name: James Braithwaite
Age: 13
Estimated Birth Year: abt 1828
Gender: Male
Where born: Cumberland, England
Civil Parish: Penrith
Hundred: Leath
County/Island: Cumberland
Country: England
Street address: Eamont Bridge
Occupation: Father Corn Miller

Thomas Braithwaite 45
Hannah Braithwaite 40
Joseph Braithwaite 20
Mary Braithwaite 15
Hannah Braithwaite 14
James Braithwaite 13
John Braithwaite 10
Jane Braithwaite 8
Thomas Braithwaite 6
Henry Braithwaite 2


1851 census
Name: Thomas Braithwaite
Age: 56
Estimated Birth Year: abt 1795
Relation: Head
Spouse's Name: Hannah Braithwaite
Gender: Male
Where born: Melmerby, Cumberland, England
Civil Parish: Cliburn
Country: England
Street address: Cliburn Mill
Occupation: Farmer

Thomas Braithwaite 56
Hannah Braithwaite 51
Joseph Braithwaite 28
John Braithwaite 19
Thomas Braithwaite 15
Jane Braithwaite 17
Henry Braithwaite 12
Agness Braithwaite 9
Watson Braithwaite 6

This could be Thomas's baptism:

England Births and Christenings
Name: Thomas Braithwaite
Gender: Male
Christening Date: 01 Jan 1796
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Birth Date: 25 Jan 1795
Father's Name: John Braithwaite
Mother's Name: Elisabeth Watson
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

Siblings:

England Births and Christenings
Name: John Braithwaite
Gender: Male
Christening Date: 07 Sep 1801
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: John Braithwaite
Mother's Name: Elizabeth Watson
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Mary Braithwaite
Gender: Female
Christening Date: 10 Dec 1804
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: John Braithwaite
Mother's Name: Elizabeth Watson
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Jane Braithwaite
Gender: Female
Christening Date: 23 Jan 1812
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: John Braithwaite
Mother's Name: Elizabeth Watson
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Elizabeth Braithwaite
Gender: Female
Christening Date: 29 Jun 1808
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: John Braithwaite
Mother's Name: Elizabeth Watson

England Births and Christenings
Name: William Braithwaite
Gender: Male
Christening Date: 14 Dec 1797
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: John Braithwaite
Mother's Name: Elizabeth Watson
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Ann Braithwaite
Gender: Female
Christening Date: 17 Jun 1793
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Birth Date: 30 Sep 1792
Father's Name: John Braithwaite
Mother's Name: Elisabeth Watson
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

England Births and Christenings
Name: Sarah Braithwaite
Gender: Female
Christening Date: 13 Feb 1790
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: John Braithwaite
Mother's Name: Elisabeth Watson
Indexing Project (Batch) Number: C06092-1 , System Origin: England-ODM , GS Film number: 90646

Sarah Braithwaite born 1790 would suggest John and Elizabeth married about 1789

I could not find the marriage

This could be John and Elizabeth which suggests they were born c1771

1841 census
Name: John Braithwaite
Age: 70
Estimated Birth Year: abt 1771
Gender: Male
Where born: Cumberland, England
Civil Parish: Melmerby
Hundred: Leath
County/Island: Cumberland
Country: England

John Braithwaite 70
Elizabeth Braithwaite 70

Found these but cannot be certain it is the John who married Elizabeth Watson

England Births and Christenings
Name: John Brathwait
Gender: Male
Christening Date: 16 Apr 1768
Christening Place: MELMERBY,CUMBERLAND,ENGLAND
Father's Name: John Brathwait
Mother's Name: Ann
Indexing Project (Batch) Number: C06092-2 , System Origin: England-ODM , GS Film number: 90646

Name: John Braithwaite
Spouse's Name: Ann Bolt
Event Date: 30 May 1757
Event Place: Melmerby,Cumberland,England
Indexing Project (Batch) Number: M06092-2 , System Origin: England-ODM , GS Film number: 90646

Thank you so much wizard28! I have a bit of reading to do now.

Hi Fiona1973,

I am the great great granddaughter of Joseph Braithwaite, Mayor of Dunedin.
I too am researching my family and have information that you may be interested in, might help to fill a few gaps. If you see this post, flick me through a reply.

Regards.