A Catholic mission began in Tynemouth 1869 in a room in the bishop's house. A church, Our Lady & St. Oswin, was built 1871. Diocese of Hexham & Newcastle.
https://www.twsitelines.info/SMR/9396Baptism, marriage & death registers from 1871 in Tyne & Wear Archives.
Tyne & Wear Archives also appears to have some earlier Catholic baptism records for Tynemouth, various runs of years, starting 1784, with long gaps. Maybe there was a visiting priest before the mission.
The map on GENUKI doesn't show any Catholic churches in Tynemouth. A search for nearby churches in a radius of 5 miles only came up with R.C. churches in South Shields, Jarrow and Hebburn. Foundation years weren't shown for any. Perhaps volunteers haven't got around to R.C. churches north of the river yet.
https://www.genuki.org.uk/big/eng/NBL/TynemouthSelect tab "Nearby churches" and adjust distance. Catholic churches locations are yellow markers on map. Enlarge map, select a marker to see name. Clicking on a name usually shows more information on another page but info for those churches is minimal.
From menu on main Tynemouth page select topic "Church Records". Go past list of Anglican churches on Church Records page to Non-conformist. Click on link for non-conformist records. This takes you to an alphabetical list. Look under T for Tynemouth R.C. baptisms. There may be other R.C. entries on the list.
When there were few chapels, few priests and a growing number of Catholics in a district as there were in many areas in England mid C.19th, practice was for a priest to travel to say Mass for congregations in whatever premises were available. Priests from different churches may have shared duties.