If you are in a tenement I guess you do not have access from above.
From what you say the whold building has unstable ceilings. The nails holding the laths up rust away over time so the ceiling is only staying up out of habit and may lose interest at any time
If it has started to sag, the broken snots above may prevent it being pushed up flat.
Having boards screwed up may do the trick. Ask neighbours who they used and see if the work seems good.
If it has to come down, you can retain cornices, coving and roses. These were (almost) invariably made of fibrous plaster in a Victorian factory and nailed into place when the ceiling was being built, and you can saw the rest of the plaster away and leave them in place. An experienced plaster restorer will know how.
Old lime plaster is very thick and heavy, and very good at blocking sound. If yours comes down, have two layers if plasterboard plus a plaster skim. You will be shocked at the noise transmission if you just have a cheap layer of plasterboard. The thick ceiling will also give better fire protection.
You can also put dense noise-reducing fibreglass batts between the joists if you have access.
The amount of dirt is staggering. Empty the room of belongings and carpet before starting, and buy or hire a canister vac and several plastic dustbins.