If a pregnant woman is 3cm and loses her mucus plug how long after will labor start?

that totally depends from person to person. Every birth and every pregnancy is different. Generally speaking though, if your waters break, your labour is a lot quicker but that is no guarantee either. I was 2 cm dilated for a month before my son was born and when my water did eventually break it took 5 hours from the time my water broke until he has born. I was in active (pain) labour for about 3 of those hours. With my first son I was in labour for 18 hours but the hospital had to break my waters. Good luck with everything.