jimrich, good evening.
Can you tell us what age is the flat?
The pictures look like the plaster is Lath and Plaster but not easy to tell?
If the ceiling is Lath and Plaster, then you may consider locating the joists, use a Magnet, stronger the better and the magnet will be attracted to the nails holding the laths to the Joists, mark the position using a pencil.
At this point, you could fix some thin straps of timber flat on to the plaster, screw the straps into the joists, or?
Using the pencil marks [put plenty of them on when using the magnet] you can simply so called overboard, that is screw fix plasterboard over the entire old ceiling screwing the plasterboard through the old Lath and Plaster into the joists.
Watch the length of the Screws, it is a two man [minimum] job
Advantages of over boarding. cheap, as you wanted. and you should not have all the fuss of the mess of the removal of the existing old damaged ceiling, removal of an old Lath and Plaster ceiling is really messy.
Finish can be a skim coat of plaster.
Hope this assists or gives you some Ideas?