How to determine if the mildew on the bathroom ceiling comes from the upper floor

Yikes, this is really the first time I have ever heard of this configuration. Could this have been part of a previous central forced hot air system? It is not common to tie exhaust vents from a bathroom to any other living space. A dedicated exhaust fan vented to outdoors with a one way damper is a very important way to remove excess moisture from bathrooms and help prevent mold and mildew growth. It is not acceptable to vent a bathroom to another living space or into an attic. It is equally not right to tie this type of vent to any other air plenum. If you are feeling cold outside air through these vents, you are getting huge heat losses. There is no need to introduce outdoor air into your house intentionally unless it is done with specifically designed air exchange equipment in an extremely airtight designed house. I think you need to determine the routing of these "phantom" vents and come up with a plan to isolate the bath vents to the out of doors with a proper bathroom fan unit and block your heat form other areas from being wasted out an open plenum.

What this could be is that you have a slow leak from the floor above. Probably a water feed rather than a drain. Although it could be from the heating system if you have a water or steam system with radiators. When you turned the water on the house it started to drip, and the ceiling soaked it all up. Eventually the ceiling became saturated and started to drip onto the floor.

If you do have a slow leak then obviously you need to find the source. It's probably a leaky pipe joint, or maybe a loose compression fitting. The only way you're going to find it is to get access, as for how that completely depends on your house's construction. There may be a panel in front of the bath that can be removed, etc.

What you need to do is meticulously go through the whole bathroom above and look for any kind of access to pipes you see, and look for signs of dripping moisture, oxidation on the pipes, or hard water deposits. As it sounds like the whole ceiling needs to come down on the floor below do that first, then use that access to find your leak.

As for how to fix it if it's a compression fitting get out your wrench and have at it. If it's a solder joint or a bad pipe get a plumber in unless you have the tools and the experience. Good luck!