Many thanks to all further replies; amondg, Treetotal and Milliepede.
In answer to your questions, the family were Catholics and we've trawled the GRO website so many times, alas to no avail. We don't have Mark's death certificate but we do have John Francis's death certificate in New York, and Mark Carroll and Catherine Fagan are shown as his parents.
The grave in the Holy Cross Cemetery, Brooklyn where Mark and John Francis are buried was actually bought by John Francis, and JF's mother Catherine is also buried there, as was his own wife Teresa.
Yes, they are the right US census records - the dates of birth for the family on these documents do tend to jump about a lot, to the extreme - seems to be a common thing back in those days - total lack of consistency in birth dates, particularly Irish Census records, as I've found with other branches of my family research. People had all sorts of personal reasons for changing their birth dates. Very frustrating (and inconsiderate

) for we modern family history researchers!
We've tried every type of spelling variation for all the names, and also searched high and low across all birth and marriage records in England (way beyond Manchester) and across all counties of Ireland.
It's a total mystery and I wonder if John Francis was never registered as a civil birth, but that his baptism lies somewhere in un-digitised Catholic church records, most likely Manchester.
One thing I've always been able to rely upon in my own Irish Catholic research is baptism records - they might've missed the odd civil registration but woe betide a child not getting baptised!
All input has been much appreciated. I guess our search goes on and hopefully more baptism records will come on-line in due course.
Runner55