I booked a car in December but as of January it is not delivered yet. Will I get cash discount offers which were there in December? or do I have to pay the price as will be in January?

Mostly the offers and prices will stay true on the day you pau the first token to company. on the same day a fixed price will be quoted for you which will not change later on

This totally depends on the brand. So, we would suggest you have a word with the dealership where you have booked the car as they will be the better person to assist you.