Well I was in your situation previous year. Let me explain it to you in a witty manner, since all three are comparable branches at one of the best NITs.

NIT Warangal Chemical

Take this if you enjoyed Chemistry and Mathematics back in school and jee days. Being located near Hyderbad and Southern Industrial Belt, you dont quite need to worry about placements. Chemical Engineering is a broad field and enables you to shift to Process Development, FMCG, Operations, Manufacturing as well as Pharmaceutical careers later in future.

NIT Trichy Metallurgy

Choose this if you want to study in the best NIT of the Country, and at the same time you liked all of Physics, Chemistry and Mathematics back in school.

Material Science and Metallurgy is both the oldest and at the same time the newest branch of Engineering. There is huge scope of research and development in Material Science in future and tbh curriculum is very structured and application oriented. If you have plans of going for a Masters in US or UK later on in future, there is no better option than NIT Trichy Metallurgy. I myself am a budding Metallurgist from IIT Kharagpur ,and trust me, you would like every part of it. Metallurgical Engineering has everything from the Physics and Mechanics you learnt in School Days to the Chemical Kinetics and Thermodynamics which you studied during JEE Preparation.

NIT Trichy Production

Choose Production Engineering at NIT Trichy only if you are interested in developing a career in Non Core/Finance/IT/Analytics Sector. I think you understand what I mean. Its curriculum is quite spread out or in simple terms relaxed and includes advanced courses like Business Analytics, Six Sigma, Production Control and Operations Research. Overall the curriculum is such that, you get a lot of time to yourself to develop and practice Software and Analytical Skills in demand.

At NIT Trichy, Placements are not quite a concern, but do choose the branch which takes you into confidence. At the end, its your career, so decide wisely.

For more Information and College Prediction do follow the link below

https://engineering.careers360.com/jee-main-college-predictor?icn=QnA&ici=qna_answer