Hello Devansh

The online application process of NID DAT and NIFT entrance exams has been completed. You cannot apply for the exams now.
NID DAT Prelims entrance exam will be conducted on 19 March 2019.

The Design Aptitude Test (DAT) is conducted by National Institute of Design for admission to the undergraduate design programme at the Ahmedabad campus and National Institute of Fashion Technology conduct NIFT entrance exam for admissions to Bachelor of Design and Bachelor of Fashion Technology programmes across NIFT campuses. 

You can apply for admission to other institutes. Symbiosis Institute of Design (SID) conducts SEED exam for admission to Design courses. 
SEED 2019 application forms will be released tentatively in the last week of January. You can go through the link below to check the details about the exam and stay updated 
https://design.careers360.com/articles/seed-application-form

Good Luck!