When will my premature baby start reaching developmental milestones on time?

That depends... It depends on how early your baby was. A premie's development is compared with others who are the same "corrected age" (chronologic age minus the number of weeks or months the baby was born early) until age two, when most will be "caught up" developmentally with their chronologic age-mates (babies born on the same date as they were). If a baby is only mildly preterm, he may catch up earlier.
The "Preemie Fudge. Factor" is used by DB Pedes & Early Intervention Programs till chronological age 2 to 2 1/2 years. Corrected age: Subtract gestational age at delivery in weeks from 40. Subtract that number from the chronological age in weeks. See http://www.healthychildren.org/English/ages-stages/baby/preemie/Pages/Corrected-Age-For-Preemies.aspx. EI prevents bad motor habits & teaches you what to work on next.
Few weeks to months. The additional time a preemie needs to reach developmental milestones varies among babies. Most preemies are only mildly delayed, and will reach their milestones a little later than normal, ranging from 1 month later to 6 months later, depending on how premature the baby was, and on which milestone is being looked at. The doctor will let the parent know if part of the development is worrisome.