In some places there was no Catholic Church until the 2nd half of the 19th century. St Godric's RC Church in Durham City, for example, was built in the 1860s with Irish money.
Prior to that some Catholics used other churches - my Irish Catholic ancestors were married in St Oswald's Anglican Church in Durham in 1847. However I've yet to find the baptism records for their two sons, John Loftus (b c 1850) and Martin Loftus (b 1851), so wonder which church they used for that.
Sadly there are very few clues to help us identify the birthplaces of our Irish Durham ancestors, so Founder's suggestion of seeking out RC parish records where they exist, is a good one.
With best wishes
Moni