A variety of birth names and dates can be found; the following entries look most likely to be hers:

This birth was found in 1851:

First name(s) AMY MILBROUGH LOUISA
Last name COX
Birth year 1851
Birth quarter 1
District Manchester
County Lancashire
Country England
Volume 20
Page 624
Record set England & Wales Births 1837-2006

This is the accompanying baptism:

First name(s) Amy Milbrough Louisa
Last name Cox
Gender Female
Birth year 1851
Birth place Manch'R
Baptism year 1851
Baptism date 17 Apr 1851
Residence Manchester, Lancashire, England
Place Manchester
County Lancashire
Country England
Father's first name(s) Thomas
Father's last name Cox
Mother's first name(s) Ann
Mother's last name Cox
Record set England Births & Baptisms 1538-1975

The sourch for the above baptism was Family Search where they have more details:

Name: Amy Milbrough Louisa Cox
Residence Place: Manchester, Lancashire, England
Gender: Female
Christening Date: 17 Apr 1851
Christening Date (Original): 17 Apr 1851
Christening Place: Collegiate, Manchester, Lancashire, England
Birthplace: Manch'r
Birth Date: 06 Feb 1851
Father's Name: Thomas Cox
Mother's Name: Ann Cox

Another entry on Family Search says the baptism entry was Manchester cathedral.

In what follows, I wonder if the Trotter principle was being applied, that no entry of birth could happen before the baptism was done, the true recognition of name and existence?
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In 1851, eventually and using the parents from the above baptism, this census entry was found which corroborates the above birth entries apart from the little matter of Amy’s birth place:  (Judging by Ann’s young age, it looks like she was Thomas’ second wife and that Henrietta was the daughter of an earlier marriage.)

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

Thomas Cox Head Married Male 44 1807 Late Publican  St Luke's, Middlesex
Ann Cox Wife Married Female 24 1827 Dress Maker  Richmond, Surrey
Henrietta Cox Daughter Unmarried Female 14 1837  -  Manchester, Lancashire
Amy Cox Daughter Unmarried Female 0 1851  -  Richmond, Surrey,
Elizabeth Smith Mother In Law Married Female  60  1791  -  Middlesex
Elizabeth Smith Sister In Law Unmarried Female  33  1818  Dress Maker  Chelsea, Middlesex,
Mary Smith Sister In Law Unmarried Female  19  1832  Dress Maker  Chelsea, Middlesex
Jane Smith Sister In Law Unmarried Female  17  1834  Dress Maker  Edinburgh, Midlothian, Scotland
Fanny Smith Niece Unmarried Female 4  1847  -  London, Middlesex
Herriet Morris Visitor Unmarried Female 18  1833  Dancer  London, Middlesex
Emma Morris Visitor Unmarried Female 16  1835  Dancer  London, Middlesex

Address & Ref:

Street Quay Street
Parish Manchester
City Manchester
County Lancashire
Archive reference HO107
Piece number 2227
Folio 282
Page 12
Record set 1851 England, Wales & Scotland Census
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In 1861 she and probably an elder sister were boarding in Dudley; might there have been a birth happening at her parents’ home?  Anyhow she is listed as at least a year younger than on the birth register:

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

Julia E Cox Lodger Unmarried Female 13 1848 -  Buckinghamshire
Amey M Cox Visitor - Female 8 1853 - Bilston, Staffordshire

Address and Ref:
Street Church Street
Parish Dudley
City or borough -
Town Dudley
County Worcestershire, Staffordshire
Archive reference RG09
Piece number 2056
Folio 23
Page 40
_________

From now on, I am not sure that I have the same person, certainly the brth and the 1851 census agree.  1861 census is possible,  But the 1871 census increasingly looks wrong and the post-marriage entries are not ocrroborated by eany birth entry.
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In 1871 this is very probably her, showing for the first time, her parents.  She is given the same birth year as in the 1861 census:

First name(s) Last name Relationship Marital status  Sex  Age  Birth year  Occupation  Birth place
Edwin Cox Head - Male 46 1825 - Oxfordshire
Mary K Cox Wife - Female 34 1837 - Lancashire
Amy M Cox Daughter - Female 18 1853 - Staffordshire
William H Cox Son - Male 9 1862 - Worcestershire
Edwin Cox Son - Male 7 1864 - Worcestershire
Alexander G Cox Son - Male 4 1867 - Worcestershire
Charles J A Cox Son - Male 1 1870 - Yorkshire

Address & Ref
Street Constitution Hill
Parish Dudley
County Worcestershire, Staffordshire
Archive reference RG10
Piece number 3012
Folio 21
Page 4
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In 1878 for the first time she was in the register as Milbrough
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In 1881, see her husband’s entry, she gave her age as 34 so born in 1847, perhaps making herself a little bit less of a child bride.  She was written as Milborough on the enumerator’s sheet and said to have been born in Melcombe Regis, Dorset which was consistently given on all following censuses.
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I cannot find her death on FreeBMD with a search for ‘Millicent Bus*ld’ died between 1882 and 1950.

But this has to be her death:

Surname First name(s) Age District Vol Page
Deaths Dec 1912   (>99%)
BUSFIELD Milbrough 65 Dorchester 5a 354
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Their marriage record:

Surname First name(s) District Vol Page
Marriages Mar 1878   (>99%)
Busfield Carrer Fothergill Ashborne 7b 681   
BUSFIELD Currer Fothergill Ashborne 7b 681   
Cox Milborough Ashborne 7b 681   
Seals John Ashborne 7b 681   
Watson Annie Ashborne 7b 681