The birth certificate I obtained for Joseph BELLAMY 1857-Deceased helped me confirm his mother to be one Emma Bellamy (née Emma BRADSHAW 1830-Deceased), born circa 1830 and at the time of Joseph's birth in 1857, living at 32 Castle Yard, Sheffield, Yorkshire.
The Census of England and Wales 1841 returns an 11 year old Emma BRADSHAW living with her parents William BRADSHAW and Elizabeth BRADSHAW in the Walkley district of Sheffield. William and Elizabeth were both born circa 1801.
Though I am once again in the realms of conjecture, my research does have an air of authority that is worth pursuing.
William BELLAMY has proved a little more difficult. My research has revealed several BELLAMYs whom are worth researching further, though at this time I cannot even propose a convincing hypothesis.