I have lost nails before. (Once when I was young, my brother shot me through the thumb with a nail gun. Lost that nail... then my sister slammed the 3 middle fingers on my right hand in a car door, lost 2 nails there... Fell and caught my index finger and popped the nail clean off...) Honestly, once the nail is gone it really doesn't hurt so bad after the first few days. The skin kind of toughens up where the nail was. To keep it clean, you can just wash it often with soap and water. Maybe put neosporin on it until it's toughened up a bit. You can put a band aid on it... but I don't think a 3 year old will keep it on. My nails usually took 2-3 months to grow back... but I have really fast growing nails, so I'm not sure...
You might want to take them in to their doctor though... if the nail bed itself is damaged, then the nail could grow in discolored or weirdly shaped. (My one thumbnail is all wavy now from damaging the nailbed...)