It sounds as if you have your cats may be suffering from an upper respiratory infection (URI) or the cat flu. This condition is somewhat similar to a cold in humans in that they are usually secondary to viruses: Herpes or Calici viruses are the two most common causes of this condition in a cat.
And, it's quite common for shelter or rescue cats to be exposed to these viruses which can then be transmitted to other cats. I believe you indicated that you have three cats so the other one may also start to show symptoms soon...or not. Just like the cold in a human, not everyone who is exposed will develop symptoms.
Secondary bacterial infections can develop (the nasal discharge can be green or yellowish in color) and when this happens, antibiotics are usually dispensed.
These episodes typically last anywhere from 10-14 days just to give you an idea of the time line.
What you can do for them at home would be the following:
1. Use warm water to clean the nose of any discharge that builds up. Over the counter artificial tears can be used to flush out the eyes should any eye issues develop.
2. Nasal decongestant drops can be helpful although many cats don’t like them very much as you can imagine. The medicated ones (a-c below) should only be used for three days in a row, with one drop in each nostril; otherwise what’s known as a rebound effect may occur. This is one reason why I prefer"d":
a. Pediatric otrivin=0.05% xylometazoline
b. “Little Noses" Decongestant Nose Drops with phenylephrine hydrochloride
c. Oxymetazoline which is the primary ingredient in Afrin or Zicam
d. "Little Noses" Saline Spray/Drops non medicated which can be used more than three days in a row.
3. You can place them in a bathroom with hot water running so that the steam can loosen up secretions although this is somewhat wasteful of water. An alternative would be to run a humidifier or boil water on your stove to moisten the air.
4. L-lysine can help if any eye issues should develop. Herpes viruses can settle in the eyes and cause issues there. This supplement would be given daily (250 mg). The feline product is available on the internet as either a chewable treat or a paste; the human version comes in a capsule and the contents can be sprinkled on wet food and given this way as another option.
5. Probiotics such as Forti Flora have been shown to help many of these cats with URI's; probiotics aren’t just useful for gastrointestinal issues. This particular product is available as a powder which is sprinkled on the food every day and can be purchased at many pet/grain stores or online.
Patients with the Feline Flu often seem to worsen before they start to show improvement so I don't want you to worry if this should happen.
Most cats will recover just fine but it may take a few more days.
I hope this helps. Deb