I have also had a frustrating time researching Henry John Fison. My link is that Henry John was my maternal great grandfather through my grandfather Sydney Cripps Fison and my mother Hilda Fison.
However, there is good evidence that Henry John Fison was in fact born Henry John Brown in 1821/22. I have found the following;
1) William Fison (b1831 d1871) son of William and Maria Fison and'brother'to Henry John left a will. The will identifies Henry John as a "brother by half blood" with the surname Brown. He was also more commonly known as John rather than Henry.
2) The above explains the 'John Brown' that appears in the 1841 census for William and Maria in Pampisford. His age is given as 15 which means 15 to 19 years old and fits the probable birth date. I also noticed that Maria's sister Hepsibah Ruffell(Russell?) is living with the family.
3)In the 1851 census he appears as John Fison age 29 years in 25 Green Street, Cambridge St Michael born Middlesex.
Can't find him in 1861 but after that he is always known as Henry John Fison on the census. He dies in 1912 age 90. In the 1911 census he tries to sign as John but this changed to Henry.
I have tried hard to link the above information to establish Henry's birth details. I have a few possibilities but nothing that fits exactly. I think his birth is between May 1821 and June 1822 based on census dates and info. and he was born in Paddington as Henry John Brown.
Hope this helps and let me know if you need any more information and details