M.Sc

Hi, you need to check with some universities, usually a three or four year degree course is required. Monash, Charles Darwin, James Cook etc. Have good nursing programs. Directly check with the universities. You may have to enroll into a higher level Bachelor's degree in nursing.

JNU is of course a good choice to do M.Sc. The university ranked 2nd in 2018 & 2017 and 3rd in 2016 among top universities in India by National Institutional Ranking Framework (MHRD).
Basically, MSc is an academic degree that's higher than bachelor's degree and lower than PhD. MSc graduates are very much in demand. You can get a respectable job in any specialisation of this field.
There are many colleges and universities that hire MSc graduates as a professor or lecturer.

St. Xavier's College, Kolkata offers M.Sc in physics, but I don't think so, astronomy is there as a specialization.

No, there is no entrance exam for M.Sc nutrition.

Hi,
There are many colleges such as Delhi University, Panjab University, JNU which have low fee structure. All you need to give entrance exam for the same of various Universities as these universities are better for M.Sc. So, you can go with these options. Good luck.

If you don't wish to pursue MS, you can go for MBA in Public Health or MBA in Hospital Management. You may also consider pursuing Doctor of Philosophy in Pedodontics and Preventive Dentistry.