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Hi!
I think you could be on the right track with this.
Nimrod married in 1854 (marriage is on freereg.org.uk)


St Edmunds Dudley 17 September 1854
Nimrod Stevenson aged 20 bachelor furnaceman, father George Stevenson horse nail maker
Margaret Griffiths aged 22 spinster, father Richard Griffiths carpenter

Birth:
Steventon, Joseph mmm Griffiths
1855 S qtr Dudley volume 06C page 2

Baptism:
Joseph Stephenton at Rowley Regis 22 March 1857
Father Nimrod, mother Margaret



As it says above, Nimrod died in 1858 and I think your Joseph might be with his grandparents in 1861

Rowley Regis
George Steventon 66
Ann ditto 66
Joseph ditto 6 (grandson)
George Longstaff 29
Dorcas ditto 27
Ephraim ditto 4
George ditto 1

I think that is a different one, she seems to be in the 1930 as the daughter of Steve Roman. So count her out ;-)

Sorry, I don't know anything much about Barbados. It is probably quite difficult for research, with records all kept locally. https://familysearch.org/wiki/en/Barbados_Genealogy

If you can get to a library with free access to Ancestry, somebody has a tree on there with quite a bit of detail about this family. Going back from Ivy's parents quite a way, though obviously everything needs checking out.

Looks like it will be fun for you to do :-)

Ivy's naturalization papers are dated 5 May 1937  and give the following

Ivy McConney
address 564 W. 149th St.
occupation hwife
date of birth 8th Oct 1889 at Hillourick BWI
Emigrated to USA from Bridgetown BWI on or about 27 Sept 1921, arrived at NY on 5th Oct 1921 on the vessel Aváre
Married to Joseph, born 19th Jan 1887 at Barbados BWI
Four children
Rosamond born 1 June 1913 NY resides with Ivy
Noel born 16 Aug 1914 NY resides BWI
Kempson born 5 Mar 1919 born NY resides BWI
Ruby born 18 Apr 1920 BWI resides BWI
Married 26 June 1926 New York
Husband naturalised 25 Sept 1919
Ivy resident in US since 5 Oct 1921

and then there is an earlier "declaration of intent"

Ivy Rosamond Phillips aged 34
Ladies maid
color white, complexion fair, height 5 feet 5 inches, weight 110 pounds, hair brown, eyes brown
Born Hillswick Village, Barbados BWI 8 Oct 1889
Now residing 2762-8th avenue
Widow
Arrived 5 October 1921

The passenger list for 5 Oct 1921 gives her father as John Noel, Hillswick Village, St Joseph
and Rosamund was traveling with her.



WW1 draft card
James H Phillips
Home address 431 W 45 st NY
date of birth 12 Feb 1888
Born Barbados BWIndies
Occupation Waiter
Employed by Lambs Club, 130 West 44 st, NY
Support wife, mother, 2 children
Married

and there is a declaration of intent (index only) giving his name as James Harfield Phillips

and a baptism

James Daniel Harfield Phillips
13 May 1888
Barbados
Father, James Bend Phillips
Mother, Amelia Grant Gibson
FHL Film No 1158014, ID 1007

The 1871 census says for District 3

part of the hamlet of Balsall north of Balsall Street - including Needles End, Wooton Green, George Road, Redfin, Meer End, Balsall Common, Holly Lane, and the Great House, Brickiln Lane, to the Grange Farm.


This map has Grange Farm and Needles End, so it must be somewhere there. A kiln is marked just south east of Needles End. http://www.british-history.ac.uk/os-1-to-10560/warwickshire/020/se