If I am considering either a CSE at BITS Pilani, chemical engineering in any IIT, or ECE or CSE at NIT Trichy, which one should I choose?

You should give preference to courses rather than the college. It is necessary to know about them beforehand. I can provide you a brief idea about each of the courses so that your choice becomes a little easier.

1. Chemical Engineering involves chemistry, physics, mathematics, and even economics. You can also check the course curriculum of the courses at some institutes such as IIT Bombay.
2. ECE is not just about circuits, semiconductor electronics, and Integrated Circuits. It involves a lot of mathematics such as calculus, functions, probabilities, and complex numbers. It also involves coding. If you are not very familiar with coding, you can do some free courses from Coursera, edX, MIT OCW, etc. before starting the course.
3. CSE will involve lots of coding. Also, the placements in any college generally tend to be better than in other branches. So you may consider CSE for that as well.

If you want to do CSE or ECE, then BITS Pilani would be an excellent choice. For Chemical Engineering, IIT Bombay is a very good choice. 

NITs generally are very good in core branches and for CSE and ECE they may not be a very good choice as the faculty is not good.