How long does it take for the bottom wisdom teeth to grow?

A long time. Tooth development and growth occurs over many years, starting out as a bud of tissue in the bone. For many people, wisdom teeth may begin erupting into the mouth during the early teen years, while others may have wisdom teeth that remain impacted due to space or position problems. Panoramic x-rays can provide information on the stage of development of your wisdom teeth.
Varies. Wisdom teeth can start erupting as early as 16 or 17 years old, and can still erupt ten or twenty years later. The time varies based on genetics and the position of the tooth buds.