I have just seen your e-mail re your connection to Bye's. Although I am not connected to your line I have been doing a virtual one-name study of Bye's for over 20 years and will be only too pleased to try to help.
Your grandmother, Winifred Bye's birth is registered at the Family Records Centre index in the December Quarter of 1885 (Ecclesall B reg Dist).
As you know, Winifred was the daughter of Henry Thomas and Clara Bye. I can tell you that Henry Thomas Bye married Clara Harwood on 4th April 1879 at St Mary, Sheffield. Henry was 23 son of Samuel Bye, Clara was 24 daughter of John Harwood (deceased).
From the 1881, 1891 and 1901 census we know that Henry and Clara had at least 7 children.
Henry Thomas Bye's birth is registered in S1856 (Clifton Reg Dist). His death is registered in M1911 (Ecclesall B Reg Dist) aged 54. He was the son of Samuel and Mary Ann Bye (nee Morgan).
Samuel Bye married Mary Ann Morgan on 23rd December 1833 (probably St John, Bristol). They had at least 8 children.
As you say, Samuel Bye was born in 1814 - I believe he was baptised on 31st January 1830 in St Philip & St Jacob, Bristol, son of Charles and Mary Bye. Samuel died in J1886 (Ecclesall B Reg Dist) aged 72.
In the 1841 census for Bristol St Philip & St Jacob (Clifton) Charles Bye (50) and Mary Bye (45) were living in Great George Street. Unfortunately, I don't have a record of the marriage of Charles and Mary but they possibly had at least 3 children - Moses (1815), Samuel (1814) and Edward (1829).
As I have record of a Moses and Elizabeth Bye in this area of Bristol having children in the early 1800's and probably earlier, they could be the parents of your Charles Bye. Moses died in 1837 aged 80. His wife, Elizabeth, I believe, died in 1844. Moses would, therefore, have been born about 1757 but further research would be required to confirm these earlier ancestors.
I have more information on your line and would be interseted in anything you can add, correct or say about your line of the Bye's. I would like to know how you are descended from Winifred Bye. Who and when did she marry, etc?