How to locate the leak in the roof

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!

I worked briefly for Sears fixing washers. A trick we used to find leaks was to put A sheet of newspaper under the machine until the leak starts. Then pull out the sheet. It will show you roughly where the leak is or where the water gets out, not always the same thing!