John Morgan Born c1814, married Bridget Co.Mayo, moved to West midlands

[Maiden Stone]

I am also interested in this thread.  My wife descends from John Morgan via his son James, and James' son George. 

What arrived in Wolverhampton was, I think, a family of three generations.  John and Bridget Morgan and their four children (Mary, James, Catherine and Bridget); John's brother Michael and his wife Mary Welch; plus Mary's father John Welch and her brother John Welch.

Do you have James or any of the others on later censuses? What information did they give about where they were born?

[Talos2000]

Do you have James or any of the others on later censuses? What information did they give about where they were born?

I have James appearing on 7 censuses:
1851:  Merridale St, Wolverhampton (St Paul);  Born: Ireland Mayo
1861:  How's Buildings, Zoar St, Wolverhampton (St Mark);  Born: Wolverhampton
1871:  116 Merridale St, Wolverhampton (St Paul);  Born: Ireland
1881:  17 Merridale St, Wolverhampton (St Paul);  Born: Wolverhampton
1891:  24 Merridale St, Wolverhampton (St Paul);  Born: Ireland
1901:  18 Pool St, Wolverhampton (St John);  Born: Wolverhampton
1911:  2 Sydney St, Wolverhampton;  Born: Ireland

[Talos2000]

....
Do you know what years baptism registers exist for that church and if there were other Catholic churches nearby? Could the family have moved parishes or new churches have been built/new missions opened during the period the children were born? This was a period of rapid expansion for the Catholic Church in England.
 ....

Sorry, no idea about that.  I'm based in Montreal, and in any case am still quite new to the genealogy game.

[Maiden Stone]

GENUKI is where I usually head first if I'm researching a family in an unfamiliar area. (UKI = UK and Ireland.) It's a volunteer-run website, organised geographically. Coverage for each area depends on how active volunteers for that area are or have been. Typically there's information about churches and cemeteries, dates and locations of registers, maps, address of archives, a brief history.  I searched GENUKI Wolverhampton.
https://www.genuki.org.uk/big/eng/STS/Wolverhampton
Substitute alternative place names and see what comes up.
Disappointingly there's no information on it for R.C. churches in Wolverhampton. The church list has C. of E. churches. The colour-coded map has only red icons for C. of E. churches, no yellow for R.C. or any other colour.
A search for Willenhall under "Place search" tab found, among other things, an old map.
Move around it to locate Wolverhampton. I enlarged it but couldn't see street names.

There are some posts about local resources at the start of RootsChat Staffordshire board. One is about R.C. registers.

Archdiocese of Birmingham Archives is presently closed and archivist is working remotely.
www.birminghamarchdiocesanarchives.org.uk/index.asp
There are links from the Resources page on the website to lists of sacramental registers on Find My Past. Look for Staffordshire on each FindMyPast list, then churches are arranged alphabetically by place. FindMyPast has: baptism registers for St. Mary, Willenhall 1864-1907; SS. Mary & John, Wolverhampton 1855-1894; SS Peter & Paul 1736-1907. There are marriage registers for some of those parishes and years on FindMyPast and a smaller collection of burial registers. (See the comment about sparse burial records for Catholics on Archdiocese Resources page.) There was also a Catholic church in Walsall, St. Mary. I didn't look at other places on the lists.   

Catholic Family History Society is on Facebook. December 2019 edition of "Catholic Ancestor", the society's journal has an article about Birmingham Diocesan Archives. (Btw a new editorial team is needed for the journal.)
Catholics living in mid-19th century England might have married in an Anglican church or been buried in an Anglican churchyard. Cemeteries were established mid-19thC. on in most places.

The family history society for West Midlands is Midland Ancestors https://midland-ancestors.uk

[Maiden Stone]

Yes, I have an Ancestry reference to a Baptismal record of 21 May 1854 for a Thomas Morgan in the church of St Peter and St Paul, which is Roman Catholic, so that would make sense.

St. Peter & Paul is situated between Paternoster Row and Ring Road St. Peter. It's the oldest Catholic church in England (according to Wikipedia) and was formerly Giffard House, home of the recusant Giffard family. Now the Catholic chaplaincy for Wolverhampton University.

[Maiden Stone]

Website for Wolverhampton archives. "The History of Wolverhampton - The City and its People"
Includes:
Article explaining about parish registers. (pdf)
Burials at Merridale Cemetery 1850-1937, A-Z surnames lists. Records of 5000 burials in "un-consecrated ground" (means not Anglican) spanning 25 years are missing.
www.wolverhamptonhistory.org.uk/resources/indexes

Staffordshire County Council Heritage & Archives
https://www.staffordshire.gov.uk/heritage-and-archives/homepage.aspx

[Maiden Stone]

Birth registrations should be on GRO births index. They may have been incorrectly transcribed or incorrectly indexed. GRO index was compiled from copies of certificates, not from originals. There are errors;

West Midlands BMD www.westmidlandsbmd.org.uk
This is another volunteer project, compiling indexes to registers of births, marriages and deaths, using original register entries as source, hopefully resulting in more accurate indexes than official GRO indexes. Unfortunately registers from only a few districts have been indexed so far and they don't include Wolverhampton.

Free BMD https://freebmd.org.uk is another source to check for those elusive birth registrations. The associated Free REG website has indexes to church registers. Both compiled by volunteers.