Answer:
No. Drastic changes in weather conditions can wreak havoc on one's immune system only if the exposure causes hypothermia, not just feeling cold.
You catch a cold by being exposed to someone who already has a cold and sneezes on you, or coughs in your face, or shakes your hand after wiping their nose, while you're in the middle of a sandwich. Or any method of transferring the viruses to your body (touching surfaces that someone with a cold recently touched, etc.).