The details in the notice show that the Trustees were paying for the re-interment in the General Cemetery and the re-erection of monuments, whether grave owners were known or not. If a family decided to re-inter in another burial ground the Trustees would also pay for that up to a stated maximum.
A very similar process was carried out in a chapel graveyard in Birmingham where two of my ancestors were buried. It was surprising to go to a municipal cemetery and find the old memorials in an enclosure arranged as they had been in the old ground - a beautiful book had been published with detailed plans, and engravings of all the monuments (the original site is now under New Street Station).
I believe the most recent removal of remains from Carver Street was for the building of a beer store for the pub. The remains were studied by archaeologists/osteologists from Sheffield University and later re-interred - not sure where.