The record shared below of a John Williams Chambers placed in a Northumberland workhome is local to the father, John Chambers who died in the Morpeth workhome.
If this is the case, then John William Chambers was placed in the Reformatory (not 'workhome') some time before 1895, when this particular John Chambers died. Do you have evidence of this?
Given that this was a place dealing with boys who were regarded as 'criminals' and 'delinquents' I think it's most unlikely that any consideration would have been given to placing him near his father.
He was certainly placed there in mid-March 1901, (i.e. very shortly before the census in which he appears there) having been found guilty of theft (see attached snip).
I have to agree with Neale that the evidence suggests that the John Chambers whose death you've found in Morpeth looks very unlikely to be the one you are looking for. If I were you I'd definitely purchase the 1894 Gateshead death certificate.
