Hello. Thank you for those helpful replies, the links to other sites particularly.
In response to particular points -
Is this your James in 1901 census? Yes, it is.
Wonder if this James is an older child who died young? That's a thought, although I also wonder if it's the same James McCullow that Gaffy found a baptism entry for on 7 September 1867, in which case it must be two different families.
Could this be the same family? That's an interesting one. The names and ages could fit but, although I don't know very much about my father's side of the family, I'd never heard mention of any ties to North Yorkshire. It's worth looking into, though.
I suspected that James and Alice McCullagh were Catholic when I saw the children listed in 1891 census. I'm not sure if you mean the ones listed in the entry for North Yorkshire, or is there one that I've missed?
At the moment, I do think it likely that the birth recorded on 6 September 1867 in Gortin, County Tyrone, is my grandfather. I still have the problem, though, of trying to prove it (perhaps I can't) and establish a link to the person I know to have been my grandfather in the later English census and marriage records. I'm grateful for the thoughts that you've given me, which may help take it forward, so thanks very much for that.
