There will be the potential for the mould to reappear as (as you have correctly surmised) there might be still some water in the ceiling space.
If you can, lift the floorboards in the upstairs room. This will allow you to see the extent of any damage and also get the air into help dry the patch out. If the area affected is not too wet then leaving the boards up for a while will help. If it is really wet then you might have to cut away the damaged board and repair the ceiling.
There are other questions here on how to patch a hole in the ceiling.
Mold goes away when the water goes away, so unless your toilet is leaking again, you have some other source of water or leak on the upper floor. This could be any number of things, even things not plumbing related, like water getting behind the siding, or a leaking roof with water running down the inside walls. As far as killing the mold that is there, I wouldn't waste my time. If you don't correct the water problem then the mold will return very shortly.
I need to figure out if there's still a leak or some kind of humidity problem. Is it safe to assume that because the problem got worse there is? Should I try to find the pipes and see if they're leaking?
Since you have had issues with the toilet in the past that would be the first area I would check. If the drain pipes for your upper floor are leaking then I would expect to see discoloration bleeding through the paint on the ceiling and the walls. If this turns up nothing then in the hole you have already cut, I would get a professional moisture detector device like the one a plumber would use.
These can help detect moisture and humidity in the air which can give you clues if you found the potential problem areas. Don't buy a cheap one because they don't work very well, I would instead see if a rental place will lend you a good one.
I think I'm seeing some black wood in there (need a better flashlight!).
This doesn't necessarily mean anything, it could be discoloration from previous water damage.
I would still like to hire someone to fix the plaster ceiling. Is it going to cost a huge amount extra if I cut through the mesh and they have to replace some of it? Does area make a massive difference in price for that kind of job? For what it's worth, I'm fine having drywall put up instead of plaster.
How big is the hole that you had cut? You can sometimes repair a hole by cutting a small piece of drywall and fitting that back in place, then using some thinset or spackle around the edges. Sand and repeat thinset or spackle two or three times until you have a smooth paintable surface.
Best Answer
If it was a toilet ring, you can be sure that it gradually started leaking longer than two weeks beforehand. It takes sometimes weeks before a slow leak accumulates enough water and softens the gypsum board enough to leak down to the ceiling below. There may very well be mould above in the ceiling space. This should be investigated asap and the appropriate authorities contacted (insurance company, condo committee, etc.) so that you are not ultimately left with the mitigation costs when it is clearly the neighbour's fault. Mould can be a health hazard. You want to be sure that there is none in the ceiling space. Good luck..