Married in 1834
30 in 1841
28 in 1851 (should be 38)
51 in 1860 when died

Name:    Robert Mathers
Gender:    Male
Birth Date:    16 Nov 1812
Baptism Date:    8 Jan 1813
Baptism Place:    Ridgway Gates Wesleyan,Great Bolton,Lancashire
Father:    Richard Mathers
Mother:    Hannah
FHL Film Number:    0560877 (RG4 1202)

Three older brothers; Richard, John and Henry all born Harwood, so probably Robert was too, most likely Raikes Farm, Harwood.

There are several possible deaths for Robert in Bolton.  However, 1860 fits well with family split up and his son John being in lodgings in 1861.

Lancashire Death indexes for the years: 1860
MATHER    Robert    51    Eastern Bolton    EB/32/283

Name:    Robert Mather
Death Date:    Jun 1860
Cemetery:    Tonge Cemetery
Burial or Cremation Place:    Bolton, Metropolitan Borough of Bolton, Greater Manchester, England
Has Bio?:    N

Name    Robert Mather
Register Type    Burial
Death Date    1860
Burial or Cremation Date    14 Jun 1860
Burial or Cremation Place    Lancashire, England

The Robert Mather that died 8 Jul 1889 at Poulton-le-Fylde, Blackpool of correct age 78 was born 7 May 1811 so is another.

Both 1841, and 1851, Mary and Robert are the same age, so Mary probably born about 1812.

Married in 1834 so assumed be over 17, born before 1817.
25 in 1841 (rounded, so born about 1816 to 1820
28 in 1851 (should be 38 ?)
33 in 1858 (when died)

Mary must have died between Q4 1851 (when Elizabeth born) and 1861 census when children are with other family members.

If born soon before 1817, then most likely the Sharples death.

LancashireBMD shows age at death and has few possible deaths for Mary Mather in Sharples (i.e. registered Bolton or Bury):
1) 1851, South Bury, aged 40
2) 1857, East Bolton, aged 53
3) 1858, Sharples, aged 33
4) 1858, East Bolton, aged 48
5) 1861, West Bolton, aged 55
6) 1862, Little Bolton, aged 56

Lancashire Death indexes for the years: 1851
MATHER    Mary    40    Bury South    BS/9/71

or
Lancashire Death indexes for the years: 1858
MATHER    Mary    33    Sharples        SH/4/493

or
Lancashire Death indexes for the years: 1857
MATHER    Mary    53    Eastern Bolton    EB/28/24

There are 6 different MATHER marrying Mary in Bolton 1844 to 1851, so  the Sharples death is quite likely another.

Not the John, son of Robert and Mary Ann.  That John was from Pendleton, baptised at Manchester and his father was a machine printer.

Born 2 Mar 1839 according to US Naturalization Record Indexes,

Aged 12 in 1851, born Bolton
Aged 22 in 1861, born Bolton
Aged 27 in 1870, born England (with Betsey 25)

Aged 24 in 1870, born England
Aged 34 in 1880, born England
Aged 50 in 1897 (when married), born England
Aged 53 in 1900, born March 1847 England
Aged 69 in 1910, born England

Lancashire Birth indexes for the years: 1839
MATHER    John    CARR    Western Bolton    WB/3/241

MATHER, JOHN        CARR 
GRO Reference: 1839  M Quarter in BOLTON UNION 
Volume 21  Page 116

Robert was 69 in 1910 and had started using his correct age again.  By 1920, he had died and Martha was head of house, Widowed.

U.S. City Directories, 1916, Massachusetts Lawrence
Mather John (Martha) died Apr 20 1916
Martha widow, John h 19 Willows.

Lancashire Birth indexes for the years: 1842
MATHER    Peter    CARR    Eastern Bolton    EB/8/198

MATHER, PETER        CARR 
GRO Reference: 1842  M Quarter in BOLTON UNION 
Volume 21  Page 107

Lancashire Death indexes for the years: 1842
MATHER    Peter    0    Eastern Bolton    EB/6/350

Name:    Peter Mather
Gender:    Male
Age:    0
Birth Date:    1842
Burial Date:    14 Aug 1842
Burial Place:    Bolton, England
Father:    Robert Mather
FHL Film Number:    1538365
Reference ID:    It.55 Pg E42/5

Aged 7 in 1851

Like her brother John, when Nancy came to the USA, she took over 10 years off her age.

Aged 50 in
Aged 51 in 1909 when died

Lancashire Birth indexes for the years: 1844
MATHER    Nancy    CARR    Sharples    SH/2/388

MATHER, NANCY        CARR 
GRO Reference: 1844  M Quarter in BOLTON UNION 
Volume 21  Page 193

Name:    Nancy Mather
Baptism Date:    7 Apr 1844
Baptism Place:    Bolton-le-Moors, St Peter, Lancashire, England
Father:    Robert Mather
Mother:    Mary Mather
Reference Number:    GB127.L243/1/3/17

Abode: Sharples (Spinner)

The Nancy Mather that died 1869 was the wife of James Mather, so NOT this death:
Lancashire Death indexes for the years: 1869
MATHER    Nancy    27    Western Bolton    WB/38/472

Instead, Wills and Probates:
Name:    Nancy Sanderson  [Nancy Mather]
Gender:    Female
Race:    White
Age:    51
Birth Date:    abt 1858
Birth Place:    England
Death Date:    8 Jul 1909
Death Place:    Lawrence, Massachusetts, USA
Father:    Robert Mather

Aged 4 in 1851

Lancashire Birth indexes for the years: 1847
MATHER    Peter    CARR    Sharples    SH/3/464

MATHER, PETER        CARR 
GRO Reference: 1847  J Quarter in BOLTON UNION 
Volume 21  Page 218

Name:    Peter Mather
Estimated birth year:    abt 1847
Registration Year:    1915
Registration Quarter:    Oct-Nov-Dec
Age at Death:    68
Registration district:    Bolton, Lancashire
Volume:    8c
Page:    532

Name    Peter Mather
Register Type    Burial
Death Date    1915
Burial or Cremation Date    9 Oct 1915
Burial or Cremation Place    Lancashire, England

Lancashire Birth indexes for the years: 1851
MATHER    Elizabeth    CARR    Western Bolton    WB/23/208

MATHER, ELIZABETH        CARR 
GRO Reference: 1851  D Quarter in BOLTON UNION
Volume 21  Page 161

Only matching death for Elizabeth Mather is in Little Hulton, 1887, aged 36.