The key is knowing when the leak started. Has this been something that has been very small and has grown slowly over years? Or is it something that just started, and got bad quite quickly. If it is the former. you probably don’t want to be taking a bath. Wood takes a while to rot, but several years could significantly weaken it.
If it is a sudden leak, then something has failed. I assume you would mention it if it was on the floor of the bathroom, so it is probably not something in the bathroom, unless you have some pipes hidden in the walls. Anyway, joints can fail and even pipes can fail fairly catastrophically. On the other hand, you don’t really say how much water is coming down.
If the leak is in the ceiling, then you’ll probably have to dig up the floor in the bathroom to get at it. I would start getting estimates for the job now.
Have you tried putting your weight on the floor of the bathroom to see how sturdy it is? You kind of press down in an almost hopping motion. If you have no movement, you do it harder. If there is movement, you probably should test how much there is, to get a sense of how far things have gone.
I seriously doubt your tub is going to crash through to the floor below. If it starts happening, you’ll feel it and be able to get to safety long before it actually falls through. My guess is that it will be fairly safe to take a bath. Probably the situation will help you save water, as you’ll use less water in the tub and take a shorter bath. Just in case.
On the other hand, I wouldn’t want to run more water through my pipes than I absolutely have to. I certainly would be leery of it if it were the waste pipe that was compromised. Bath water is bad enough to come down, but the waste from the toilet? Not interested. But probably the leak is on the supply side. Good luck!