It should not take long if they know what they are doing. The costs of home repair materials has stayed fairly constant so they should not have to guess at the price or have to do much running around to check prices, unless there is some sort of special problem. I have had contractors give repair estimates right after looking at the job. It should not take more than one day.
Water damage to a popcorn style ceiling can mean some major tear out work depending on how much damage was done to the ceiling material and how the water damage happened. We had some rood damage due to a hurricane and then the water leaked in and damaged the sheet rock ceilings with a popcorn finish. If the sheet rock has been damaged, ie: it is split, sagging, or has large holes, then it will need to be replaced. First make sure that where ever the water came from is fixed, such as leaky pumbing or a bad roof. You do not want to fix the ceiling only to have it ruined again down the road. It is easier to have the popcorn finish removed rather than try and replace it, in an already built house. They have to spray on the popcorn coating if a large area has to be covered and that will mean a big mess on the walls and everthing. We simply had it removed and went for a smoother finish on the new sheet rock.
If the area to be fixed is small you can get some spray on popcorn finish that comes in a spray can. It can be hard to match the existing finish though. It is impossible to paint popcorn ceilings as the stuff will fall and come off as you try to paint it.

Water can cause mold. It may need to be replaced.

Popcorn ceilings have gone out of favor. They hold dust and it is almost impossible to match the cottage cheese curd lump size. Fortunatly they come off pretty easily. The first thing you want to do is to make sure that this isn't a pre 1980 ceiling or you may be looking at asbestos. If you are unsure then have it tested by your town or a local testing facility. If you are sure you are not dealing with asbestos then you can remove it yourself.

To remove it you need the following:

A dustpan
A spray bottle
A scraper
A large barrel lined with a plastic bag
Fabric softener

Put 1/2 cap of fabric softener in a spray bottle with HOT water.
Afix the scraper to the handle of the dustpan.
Work manageable sections at a time with the spray and let it set up for a few minutes
Scrap and empty the pan.. the scraper attached to the pan keeps most of the popcorn in the shovel and eliminates alot of cleanup.

It takes a good 30 minutes to remove popcorn in an average size room. Afterwhich you may need to patch some areas..prime and repaint. If the water damaged area has had the issue repaired you can usually get away with using the primer to cover any demarcation in color.

Finding the best qualified company or person is key here. If you have not yet repaired the source of the leak...is it the roof? or a plumbing issue?... then this is where you should start with any repair. Once the leak has been fixed, you need to evaluate the material on the ceiling.
This should be gypsum board / drywall. Just touch or poke the material and if it crumbles, or is 'spongy' you will need to replace the area of failure or the entire ceiling. You can scrape off the "cottage cheese" style ceiling with a water solution if necessary. This a rather big and messy job!
When you have done the initial leak repair and checked the material on the ceiling to be a solid material, you can move onto putting back the textured ceiling and paint as desired.

The first thing to do is to find out what is leaking. Otherwise if you fix it it will get wet again. Try to get a look at the roof to see if it is leaking. If the roof is leaking the rafters may be wet or show stains from the leak. If so, get a bid to repair the roof. Also, a pipe or fitting could be leaking. Prime suspect a leaking toilet seal. Look around the toilet base for wet spots. Check out all the sinks and tubs. Once the leak has been stopped you can move to the ceiling. With the leak stopped the ceiling will probably dry out on its own so unless the wood underneath is rotted (push on the ceiling). If there isn't a lot of "give" it should be fine. I would suggest that at a minimum you paint the ceiling. You could scrape off the "popcorn" but it is a mess to clean up. Also, in some homes there is asbestos in these ceilings making them hazardous to simple scrape off. You can take a sample to your county health department and get it tested for asbestos. If it has asbestos in it you could paint over it which would seal it up. Then you could re-sheetrock over it so it won't be disturbed.