What can/should we do to control moisture until the plumber can get here?

Mold is only a concern if an area is repeatedly soaked. If an area of the house gets wet but then has a chance to dry then there will not likely be any mold. However, given that water got under the tiles during showers there may be a mold issue in that area (and there may not be). If water had gotten under the tiles from just your bucket trick then I would not be concerned about damage, but regular showers may have created a problem. A wet and dry shop vac is good for pulling water out of carpets.