Paul - welcome to the forums.
As you have discovered 1872 falls after the period for which we have local online transcripts of baptisms. I do have access to the parish's baptisms going up to around 1908 and there is no record of a John Hill being baptised in Appleby around 1872.
But at least the date is after the start of civil registration in England.
I have found a birth index for a John Hill, registered in Q2 1872 in Ashby de la Zouch (the local registration office for Appleby Magna), it is listed as volume 7a, page 83.
Unfortunately I don't currently have an active subscription with any genealogical service that would allow me to look at the birth certificate.
If it is of any help I have found these "extra" records in the Appleby Registers for John Hill and his wife Rose Ellen ....... if this is the right John Hill.
A bit of a mystery with this first entry - Mary Hill buried 30/06/1895 - no parents listed. Mary was aged 12 days at the burial, so I guess there is a chance that a date could have been mis-transcribed somewhere, as there is a entry for a Mary Hill being baptised a month later.
27/07/1895 baptism, a daughter, Mary - John listed as a labourer.
06/12/1896 baptism, a daughter, Cecilia May - John now listed as a coal miner.
20/01/1899 baptism, a son, John Harold.
30/09/1900 baptism, a daughter, Florence Irene.
22/12/1900 burial, Florence Irene.
20/03/1902 baptism, a son, Amos Dennis.
24/12/1902 burial, Amos Dennis.
That seems to be all the Hill records on the "extra" records I have for after 1857.
The only other record I could find that may be of interest, although this could be clutching at straws.
04/10/1856 a Joseph Hill married Mary Roland - her father's name was William.
So a late 1856 marriage could be possible for an early 1857 birth of John (a bit of a rushed marriage ?) and William Roland would be the grandfather.