Very interesting place to visit, Vagrant cells are a piece of 19th century history; with the poor law in effect, workhouses were popping up all over the country and the cells were a place where the penniless could go to sleep as sleeping on the streets was in fact illegal back then.
The cells had been attached to the workhouse itself that was constructed in 1836 and the Infirmary which was built slightly later in 1869. The entire site later became a hospital which is know partly live and partly derelict. The cells itself had been put forward for listing status in 2001 but had been denied.
I haven’t found any research of the morgue itself, I would imagine it would have been there from the infirmary and early hospital days, as it was such a small place and looks to have been unused for quite a few years!