My knowledge of the Catholic Church in Sheffield is limited.
I am using the Small Guide to Nonconformist & Roman Catholic Churches a S&DFHS publication.
Not many RC churches at all till the middle of the 1850's. Sheffield's RC population would have been very small probably until the influx of Irish settlers.
Statement for the Booklet.
"The Cathedral Church of St Marie, Norfolk Row, Sheffield
Originally the Mass center in the House of the steward of the Duke of Norfolk in Coal Pit Lane (now Cambridge St) from c1700 and later at the Lords House, Norfolk Row. The mission was founded in 1728 and the chapel in 1816. The present Church was built between 1847-1850.
It became the Cathedral Church of Hallam Diocese 1980.
Baptisms 1783-1828
Births & Baptisms 1827-1840
Original Registers held at the TNA i.e. The Public Records Office National Archives.
Apart from the above Mass center there would have been a few landed gentry who had private chapels i.e. Nethergate Hall at Stannington. It was the domestic chapel of the Revell family 1742-1828 reopened in 1855 closed in 1929.
Earlier there would have "roving priests".
Catholic Emancipation occurred around the 1830's. Just Google it and you will have some of your questions answered.
Read somewhere that the Shrewsbury Chapel in tthe Cathedral the then parish church was used by Catholics. Could a baptism if performed there be registered in the records for the parish church. Shot in the dark!