Child not walking at 15 months?

I am a concerned parent. I have a 15 month old son he is extremely active, love to play with toys, uses his hands to grasps objects and always getting into something. He is a very fast crawler, and very strong. He says bye, momma, papa, and loves to wave goodbye and holds his hand to his ear to pretend he is talking on the phone. In my mind I thought he was a normal active toddler. We took him to the doctor for check up we told him he is still not walking. He will stand up when leaning on a wall or furniture but seems to not stand by himself with proper balance. When he does stand, he only stands on his tippy toes. We thought it was normal due to him using a jumper and Walker and us having a small apartment to not really be active and walk around. Doctor told us he could have cerebral palsy. That just worried me to death he seems healthy and normal and interacts with others. He will not walk and won't stand by himself but will crawl just fine and stand up holding on to furniture. Is he just a late bloomer or does he really have this condition. Doctor only gave us a month, of he walks then awesome if not she said it is cerebral palsy. He was born a month early. Should I be worried? Advice needed please