Hello, and welcome to one of the most frustrating surname searches...
From the information in your post, I think I can provide some help. John Beilby's birth was recorded in the Flamborough parish register, christened 7 Feb 1847. He is on the IGI as "John Beilby Elliot". Although I have not viewed this register as yet, it would appear that he is recorded with only his mother's name (as Mary Elliot). This would most likely mean he was born after Henry's death, but this then raises a question as to his father - Henry could have died some time between conception and birth (which seems to have happened often), or it could be that Mary was already widowed but fallen pregnant to someone else - possibly a "Beilby" (there are a couple of Beilby families in Flamborough at the time). The only hint to this could be to go through the burial register for Flamborough and see if Henry's burial is recorded within the months prior to John's birth. Although this doesnt prove him as the right father, if his burial is more than nine months prior to John's birth, it would rule him out.
I have two sisters on record for John, and he was the youngest child, being christened a little over two years after his sister Ann, so either scenario is possible for his father.
As for Henry, I have him as christened 16 Feb 1815 in Nafferton, Yorkshire, youngest child of Ephraim Ellyard and Frances Bott. Ephraim is a difficult one to trace, since although he is beleived to have been born c1781, he was not christened until 23 Jul 1809 (in Nafferton). I have so far no possible reason for his adult baptism.
Ephraim is recorded in the Nafferton baptism register as born 1781, with parents listed as Robert Ellyard and Alice. Robert Elyard is known to have married twice (Alice Robson being his second wife, his first was Mary Wyles), and also to have fathered a child to a third woman, Elizabeth Usher.
Robert Ellyard was christened 13 Dec 1745 in Nafferton (recorded as Elliott), the second son of John Ellyard (Elliott) and Ann Medforth.
This is where the line ends for now - so far, this John's origins are uncertain. He is beleived to have been born circa 1720, but as for where or who his parents are, unknown at this stage.
My involvement with the Yorkshire lines is not a direct one - I am descended from a "single-L" Elyard line who came to Australia in 1821, from London. However, my ancestors also spent time circa1700 serving in the military in Northumberland before moving back south to London. It is my current opinion that there may be a link between the two lines around the late 1600s-early 1700s, but currently this link is not able to be substantiated beyond a few speculative hints.
I do have a family of Elliots in the Berwick and Tynemouth areas who may be a part of such a link. Your ancestor Ephraim had a brother, Enoch - in the Northumberland families, the name Enoch was found in an Elliot family as the name for three seperate births - it appears that the first two died in infancy. The strong determination to use the name for one of their children hints at it being of some importance to the family there. Also it is not a common name for the time, so perhaps John of Nafferton could be from this line.
Other hints to this being part of the same family are the fact that there is a marriage of these Northumberland Elliots (curiously, another Enoch) and a Robson family - Enoch Elliot married Ann Robson in 1726 Tynemouth. It could be that the two families were known to each other for several generations.
My ancestor in Northumberland stands out since he appears to be an only child, Henry Elyard, born 1697 in Berwick. His father was noted as a soldier, and this family does move between parishes a bit in the area, so it is quite likely that there are other siblings which I have not yet located - possibly one being the father of John of Nafferton.
You should keep in mind that my research on the Yorkshire lines is still a little unconfirmed - I have only viewed a few source documents personally for this area, much of my information has come from David Ellyard, and also from two other descendents of this line still living in the Leeds area, Rowland Ellyard and Suzanne McKay (nee Tindall).
Hopefully this will give you a place to start. David Ellyard will probably reply to your post as well at some point, and he may be able to provide some additional info.