Aged 35 in 1841

There is no baptism for a Jacob Mather, anywhere in the country!
He would probably be baptised at Moor Lane, since he lived there.

Widower in 1851, aged 49
Married in 1861, aged 58, living in Staffordshire

Widower in 1851, aged 49
Married in 1861, aged 58, living in Staffordshire

Sons James and William (aged 1 and 5) were both buried at St. Peters, but Sarah and Jacob were buried elsewhere.

Died after 1861, probably before 1971.  Only 4 deaths in England FreeBMD 1861-99, and one with no age given, so is possibly this Jacob Mather:

MATHER, JACOB        64 
GRO Reference: 1864  S Quarter in LIVERPOOL 
Volume 08B  Page 209

If 38 when Sarah died, she was born about 1805

Lancashire Death indexes for the years: 1843
MATHER Sarah 38 Turton Bolton TUR/2/207

Sons James and William (aged 1 and 5) were both buried at St. Peters, but Sarah and Jacob were buried elsewhere.

John was 15 in 1841

Name:    John Mather
Baptism Date:    17 Apr 1825
Parish:    Deane, Lancashire, England
Father:    Jacob Mather
Mother:    Sarah Mather
Register Type:    Bishop's Transcripts
Reference Number:    Drm/2/72
Does the John Mather in this record match the person

Baptism: 17 Apr 1825 St Mary the Virgin, Deane, Lancashire, England
John Mather - Son of Jacob Mather & Sarah
    Abode: Great Bolton
    Occupation: Weaver
    Baptised by: Fras. Whalley Curate
    Register: Baptisms 1822 - 1833, Page 93, Entry 741
    Source: LDS Family Search

Baptism: 17 Feb 1828 St Mary the Virgin, Deane, Lancashire, England
William Mather - Son of Jacob Mather & Sarah
    Abode: Great Bolton
    Occupation: Warper
    Baptised by: J S Baker Curate
    Register: Baptisms 1822 - 1833, Page 168, Entry 1341
    Source: LDS Family Search

Name: William Mather 
Gender: Male 
Age: 5
Burial Date: 29 Apr 1832 
Burial Place: Bolton-Le-Moors, Lancashire, England 
Father's Name: Jacob Mather 
Indexing Project (Batch) Number: I01724-9 
System Origin: England-EASy 
GS Film number: 1966470 
Reference ID: p 55 #433 

James and William (aged 1 and 5) were both buried at St. Peters.

Probably born about November, as was aged 1 when died.

Baptism: 5 Dec 1830 St Peter, Bolton, Lancashire, England
James Mather - Son of Jacob Mather & Sarah
    Abode: Great Bolton
    Occupation: Warper
    Baptised by: I Jenkins Trinity
    Register: Baptisms 1829 - 1831, Page 234, Entry 1872
    Source: LDS Family Search

Name: James Mather 
Gender: Male 
Burial Date: 02 Mar 1832 
Burial Place: Bolton-Le-Moors, Lancashire, England 
Age: 0 (see other record)
Birth Date: 1832 
Marital Status: Single 
Father's Name: Jacob Mather 
Indexing Project (Batch) Number: B01480-3 
System Origin: England-EASy 
GS Film number: 1545451 

and

Name: James Mather 
Gender: Male 
Burial Date: 02 Mar 1832 
Burial Place: Bolton, Lemoors Parish, Lancaster, Pennsylvania
(Should be Bolton-le-Moores, Lancashire)
Age: 1 
Birth Date: 1831 
Father's Name: ... Mather 
Indexing Project (Batch) Number: B02938-9 
System Origin: England-EASy 
GS Film number: 559184 
Reference ID: Pg 30 

James and William (aged 1 and 5) were both buried at St. Peters.