Name from birth search for daughter on New Zealand BDM website in March 2022:
1856/2167. Bayly, Emma. Mother: Grace. Father: James

Kim Cripps (wife of Douglas Cripps, with Douglas being a cousin of Brian of nzolivers) alerted me to a Facebook post of 22 December 2023 by parliamentary MP, Andrew Bayly. Andrew wrote, "The Bayly’s arrived in New Plymouth on the ship Amelia Thompson in 1841 and I grew up on a farm near Whanganui".
Now, the Olivers had arrived on the same ship.
However, of equal interest, Brian of nzolivers maternal step-grandmother was Elinor May Gower. Elinor’s mother was Emma Bayly and Emma’s parent’s and older brother are listed on the Amelia Thompson Passenger List, along with two other Bayly familes:
1. James age 25 [Born circa 1816] and Grace age 22 [Born circa 1819] [Emma Bayly’s parents and Elinor May Gower’s grandparents] with son Arthur age 1 year
2. William age 30 and Elizabeth 26 with children Daniel 6, Ann 2, and Jane 11 months
3. Thomas Bayly age 36 and Susan Bayly age 35 with children Elizabeth 10, Thomas 8, William 6, Ann 4 and Issac 11 months

Exact date of birth, place of birth, date and place of marriage, date and place of death “James [now a widow] lived with his son Albert in the years prior to his death 30 Jun 1901 at Omata”, from:
https://winsomegriffin.com/Newsham/JamesBayly.html

Headstone at Te Henui Cemetery, New Plymouth for Grace and James Bayly and their son-in-law Daniel Berry. Added here in January 2024. Inscription reads:
To the memory of
GRACE
The beloved wife of
JAMES BAYLY
Died 5? July 1891
Aged 75 years
“The blood of Jesus Christ God’s son cleanest us from all sin”
Also her loved husband
JAMES
Died 2nd July 1901
Aged 84 years
And their son-in-law
DANIEL BERRY
Died 17th May 1917
Aged 69 years

Photos courtesy of Winsome Griffin as per authorisation in email dated 25 December 2023 and added here in January 2024.

Winsome Griffin's (nee Guilford) husband, Neville Griffin, is related to the Bayly family. Neville's maternal grandfather, George Newsham, had a sister, Emma Eliza Newsham. She married Albert George Bayly. Albert had a sister, Emma Bayly, and she was the mother of Elinor May Gower, the maternal step-grandmother of Brian of nzolivers.

And, Winsome Griffin (nee Guilford) is the granddaughter of Francis Charles Guilford (1876-1961) whose second wife, Laura Elizabeth Wood, was first married to Charles Ernest Oliver, grandson of James (1805-1886) and Ruth (nee Batten) Oliver.

Record of marriage from the Devon, Church of England Marriage records found in January 2024:
Groom: James Bayly
Groom Marriage Age: Full Age
Groom Religion: Anglican
Groom Residence Place: Fernhill
Groom Father: John Bayly
Bride: Grace Metherell
Bride Marriage Age: Full Age
Bride Residence Place: Horslett Lane
Bride Religion: Anglican
Bride Father: Roger Metherell
Marriage Month/Year: June 1839
Marriage Place: Clawton, Devon, England