How long does it take for breast buds to grow?

Varies. Breast buds are one of the first signs of puberty. Usually the time from start to finish in puberty is about 3 - 4 years. This can be prolonged though in people with lower estrogen. Girls under physiologic stress would see this. Very thin and "over exercised" would fit this category. Puberty then often starts late and takes longer. High levels of exercise in girls is not so positive!