Engineering Admission Process

No NSIT does not provide lateral entry.

LEET stands for Lateral Entry Entrance Exam. This exam is for diploma students to take admission into Bachelor's degree. Yes it affects the admission cause on the basis of this exam an individual will get admission into the college.

There is no difference between B.Tech and B.E. as both is engineering degree just a name difference.
MS Ramaiah offers B.E. not B.Tech but no problem its engineering degree.
Check College details in the link provided:
https://www.shiksha.com/college/m-s-ramaiah-institute-of-management-m-s-ramaiah-bangalore-30829

Engineering Colleges/Universities offering LEET admissions are
* Punjab University - Chandigarh
* IP University - Delhi
* Delhi Engineering College - Delhi
* Punjab University - Patiala
* KIITS University
* Thapar Institute of Engineering and Technology - Patiala
* Government College of Engineering - Pune
* Punjab Technical University - Jalandhar
* Guru Nanak Dev Engineering College - Ludhiana
* Sant Longoval Institute of Engineering and Technology - Sangrur
* Anna University - Tamilnadu , etc

DTU offers lateral entry and there are universities like Indraprastha University in Delhi NCR.

To get lateral entry in the B.Tech you need to clear your diploma of 3 years with at least 60% marks the colleges that offer lateral entry in the engineering are as follows:
Lovely Professional University-Jalandhar,
Adamas University-West Bengal,Techno India University-West Bengal,JIS University-West Bengal,Acharya Institute of technology-Bangalore,
BBIT-West Bengal, etc.

You can't get admission in NIT through lateral entry because they take students only via JEE Mains.