how long after sex does it take to induce labor?

Sex doesn't always induce labor, it really depends on your body and if the baby is ready.  If you are already dilating or effacing, then you probably have a greater chance of going into labor, but sex is not a guarantee to get it going.  I've seen on here a bunch that nipple stimulation works too, but I can tell you we tried everything under the sun with our 2nd son and I was 4cm for 3 weeks before he was born and he still came 2 days after his due date.  Sorry, I know that's probably not what you wanted to hear, but here is no science to it, it's all a matter of when your body is ready.