Birth control pills
How long does it take for your cycle to stop once you start taking the pill
It takes about 28 days or less as birth control pills take after a regular 28-day monthly cycle. This will vary per person due to body differences. There is more information on this at http://www.mayoclinic.com/health/birth-control-pill/WO00098.
About a month or two just so your cycle can adapt to the pill and how it works.
Wait one complete cycle before stopping your current birth control. It will take that long before it becomes effective. Call your Gynecologist with questions.