If you put floors in the way, then no, the bucket can't lower. You'd need to go in and channel out underneath the well for each floor. I forget how deep a well can reach, however.
Also, I doubt you'll need all that much water to begin with, especially if your basin will still be connected to the river. Whenever winter rolls around, it won't feed the basin, but any other season the water will still flow.
In fact, it's likely that the pressure will flood your fortress unless you add a floodgate or pressure relieving system.
Ah, so if I use channel on every section on every floor that will open the whole thing up? Perfect, thanks. And yeah, how many floors I need to dig is going to take a little experimenting, but I'll figure that out. I've been thinking about the pressure and I can't see much that I'd need to do, the well itself will be a floor above the river's natural one, I already tried a test well using staircases to let the water in to the lower floors as I said in my first post and sealed off the top floor with a wall, seemed to be okay.