There are no such specific prerequisites. You can develop an interest in coding within one year if you are interested.

The programming courses in the first year entail the comprehensive usage of C. It definitely is an extensive course and will take up time. You will learn the basics of it in the first year. The course includes parts like Functions, Pointers, Linked Lists, Data Structures, Object-Oriented Programming, etc.

Although, having knowledge of Python helps. If, however, you've arrived with no knowledge at all of what that is, then it is fine too. Classroom teaching does not help that much for learning to code. Mostly, you will need to learn the aspects of this language with your own efforts.

If you have any spare time, learning basic front-end web development will assist in the long run as well, like HTML, CSS, JQuery, and Javascript (If possible). This will help explore a variety of other opportunities outside your curriculum.

IIIT Allahabad is the institute of national importance, whereas Thapar Institute of Engineering & Technology is a deemed university in Punjab. The best institute among the two for pursuing engineering is IIIT Allahabad. It is far better than Thapar. Here is a brief overview of the institutes.

• The tag of IIIT is an advantage. IIIT Allahabad is not less than old IITs as it is one among the Tier-1 colleges for engineering.
• The quality of research and reputation it has in the industry makes it a lucrative choice.
• IIIT Allahabad offers only two specializations, IT and ECE. It diverts the complete focus on these two branches.
• The faculty of the institute is skilled in teaching and has vast experience in the industry. They use the industry-specific teaching approach and prepare the students to take up real-world challenges.
• The coding culture of the institute is renowned in the country. It has given birth to many excellent programmers in the world. Every year, a team goes to compete at the international finals in ACM ICPC. Heard of ‘Humblefool’? Till date, he is considered the best programmer from India and he belongs to IIIT Allahabad.

Overall, the above points conclude that IIIT Allahabad is not only the brand, but it also gives the best return on investment.

Firstly, you should know that there is no difference in the facilities and reputation of the colleges, Although BITS ranks better as an engineering school. BITS Pilani ranks #30 on NIRF's Engineering rank list and IIIT Allahabad ranks #103 on the same. But you should focus on your monetary capacity in consonance with your future plans before applying. 

My cousin studied at IIIT A and the following points are based on his experience:

• IIIT A is an excellent place to study coding. The classes are good although the syllabus has to be maneuvered because it's tough. You can manage it through attention in the classroom.
• Attendance is an important part of the academic process here because IIIT A has a 75% mandatory attendance rule.
• Without a doubt, BITS has a higher fee range as compared to IIIT A, about 7 LPA for a 4-year program.

Here's a table comparing both institutes:

Lastly, if you love the mechanical stream and are truly motivated to study Maths and Physics then you must go for it. Mechanical is a vast field as well because you'd learn everything related to mechanical processes and even production scheduling. So clearly check your interests and goals before deciding.