Hi dear, yes Bangalore weather in general is cold in the morning throughout the year almost. So such problems are common here, do not worry.
Next Steps
Try to keep yourself warm. If going out early, cover your ears, nose n mouth, fingers n toes too if possible. Physical exercises also help a lot.
Health Tips
Drink hot water whenever you can, do steam inhalation, consume lots of fresh n dry fruits n lots of raw vegetables as in salads to improve immunity.