Did 19th Century Irish immigrants use CoE churches?

[Maiden Stone]

So is it possible that a Catholic woman might marry a CoI man and then "become" CoI and baptise their children in CoE churches?! The time frame I'm looking at is marriage in 1828 (which was in Corfu no idea of denomination) and then their children were born 1838-1843 in England?

Anything is possible.
A spouse might convert or might not. A spouse might convert later in the marriage.
 If the parents belonged to different denominations they might decide which church they had a child baptised at, depending on a variety of factors. These may include: which spouse felt more strongly about their religion; opinion of their respective families; whether the spouses were equal partners in the marriage and discussed decisions beforehand or whether one spouse (or their family) was dominant; circumstances of each baby's birth, e.g. where, when, which family members were around, the community a child was born into; was a clergyman of choice available; potential future  education & employment prospects for a son; whether one side of a family had money, property or influence.

Marriage in Corfu complicates your research. Why were they in Corfu? What was his occupation?
If they had married in England in 1828 the wedding would have been in an Anglican church, regardless of their denomination (unless they were Jews or Quakers). If they had married in Ireland in 1828 their denomination(s) would have affected their choice of wedding venue.
Were there no surviving children born 1828-1837?

Unless you find baptisms for your 4x great-grandparents you can't know which denomination their parents were. Surviving R.C. registers for Balla are too late. Have you looked at the list of C. of I. registers ?
   

[Maiden Stone]

Have you looked at the list of C. of I. registers ?
 

Church of Ireland parish registers list.
 https://www.ireland.anglican.org/about/rcb-library/list-of-parish-registers
Colour-coded list arranged A-Z by name of parish. See parishes of Balla, Castlebar and Rosslee. Diocese is Tuam.
Balla: existing registers, baptisms from 1856, marriages from 1871, burials from 1870.
Castlebar: existing registers all from 1840's.
Rosslee: No dates on list. Colour-coded white which indicates local custody.

There was no C. of I. church in Balla when Samuel Lewis published his  "Topographical Dictionary" in 1837. Services were held occasionally in the courthouse. Mentions vicarages of Rosslee and Minola. Tuam diocese.

[Burto]

So is it possible that a Catholic woman might marry a CoI man and then "become" CoI and baptise their children in CoE churches?! The time frame I'm looking at is marriage in 1828 (which was in Corfu no idea of denomination) and then their children were born 1838-1843 in England?

Anything is possible.
A spouse might convert or might not. A spouse might convert later in the marriage.
 If the parents belonged to different denominations they might decide which church they had a child baptised at, depending on a variety of factors. These may include: which spouse felt more strongly about their religion; opinion of their respective families; whether the spouses were equal partners in the marriage and discussed decisions beforehand or whether one spouse (or their family) was dominant; circumstances of each baby's birth, e.g. where, when, which family members were around, the community a child was born into; was a clergyman of choice available; potential future  education & employment prospects for a son; whether one side of a family had money, property or influence.

Marriage in Corfu complicates your research. Why were they in Corfu? What was his occupation?
If they had married in England in 1828 the wedding would have been in an Anglican church, regardless of their denomination (unless they were Jews or Quakers). If they had married in Ireland in 1828 their denomination(s) would have affected their choice of wedding venue.
Were there no surviving children born 1828-1837?

Unless you find baptisms for your 4x great-grandparents you can't know which denomination their parents were. Surviving R.C. registers for Balla are too late. Have you looked at the list of C. of I. registers ?
 

They married in Corfu because John was serving in the army there. I presume that Catherine was maybe a soldier's daughter.

[Maiden Stone]

Depends on where near Birmingham they settled however there were RC/non conformist churches https://www.genuki.org.uk/big/eng/WAR/Birmingham/nonconform

Coventry, Hampton in Arden and finally Birmingham

Location of churches in Coventry area.
 https://www.genuki.org.uk/big/eng/WAR/Coventry
Select "Nearby churches" tab + distance to see map showing locations of churches, colour-coded by denomination. Yellow icon for  R.C. churches. Can select a denomination. Enlarge map. Click on a yellow icon for church name + perhaps foundation year. Click on name box for more info on GENUKI about the church. A congregation may have pre-dated foundation of a church in an area, especially R.C. and Non-Conformist.

St. Osberg's Church, Coventry founded 1843-5. (This was when the church of St. Osberg/Osburg was built. R.C. congregation in Coventry was almost a century older.)
 A Catholic congregation was founded between 1757-1760. A priest from Baddesley visited monthly to hold a service. (Baddesley Clinton was owned by a Catholic family, Ferrars). Catholics in Coventry previously attended a chapel at Wappenbury, near Leamington Spa, (later St. Anne), which was operating in 1734 and probably before. Lord Clifford, Lord of the Manor of Wappenbury was R.C. as were many Warwickshire gentry.
Services were held in people's houses in Coventry, most often in Little Park Street, 1775-1806.
Land at Hill St. was bought as a site for a chapel (St. Lawrence & St. Mary) in 1806.
Rev. William Ullathorne arrived in Coventry 1841 and began a replacement church. Foundation stone laid 1841. Rev. Ullathorne launched a public appeal and began preaching for funds, concentrating on districts where Irish labourers congregated. The Church of The Most Holy Sacrament and St. Osburg was consecrated 1844. (Personal interest here. William Bernard Ullathorne may have been a relative of a member of my ancient English Catholic family. He became the first Catholic Bishop and later Archbishop of Birmingham.)

"A Centenary Memorial of Saint Osburg's, Coventry 1845-1945" by Sebastian Simpson
"The History of The Most Holy Sacrament and St. Osburg, Coventry" by Gary Byrne
"From Cabin Boy to Archbishop", autobiography of William Bernard Ullathorne


 
 
 

[Maiden Stone]

Depends on where near Birmingham they settled however there were RC/non conformist churches https://www.genuki.org.uk/big/eng/WAR/Birmingham/nonconform

Coventry, Hampton in Arden and finally Birmingham

Hampton-in-Arden is in Solihull district.
Catholic churches in the area
https://www.genuki.org.uk/big/eng/WAR/Solihull

Holy Ghost and Mary Immaculate, Olton Priory, Olton, founded 1820's

St. Augustine, Solihull founded 1761
A piece of land was given for a chapel and priest's house in 1760. New church built 1839. (Oldest Pugin church)
https://www.staugustinesolihull.org.uk/welcome/church-building/history-of-st-augustine-church/