Ok, I was worried that we were on the total wrong path.
Between 1855 and 1859 there were 5 male Callows registered in Bham.
Joseph Dec 1857, son of James Callow & Mary Copper (I wrongly said William & Martha earlier).
William Dec 1855, again son of James & Mary.
William Sep 1855, son of William Callow & Martha Peters, he appears to die young because...
William Joseph Dec 1856, again son of William & Martha, he is listed sometimes as William, sometimes Joseph.
Which leaves Joseph Sep 1856, who appears to be the son of Joseph Callow & Emma Jones. Apart from his birth registration & baptism not sure what happened to him or his parents. There are no other Callow/Jones children registered locally.
The options seem to be that...
Your Joseph was the son of William & Eliza and he wasn't registered, not uncommon but a little surprising as they registered all the other children.
He was not their son and he was 'adopted', again not uncommon.
No way of knowing which, unless Joseph has descendants and dna can do it.
I was looking for a connection between Joseph, the father of Joseph above and your William (brothers maybe) but nothing certain.