B.Arch

Yes you can get admission in private colleges in B.Tech, but some of them have different exams like VIT, BITS, SRM, etc. So you would be required to submit the application forms. Keep checking their last dates for further details.

But there are better colleges than BGS in that budget.
The fees structure for BGS is 22 lakhs.
While colleges like Acharya Institute of Technology is offering it for 14.50 lakhs.

Just make sure that the margin between cutoff and the marks scored by you should be good and the percentage in 12th must be greater than or equal to 75. Just do hard work.
You will definitely get a seat.

You should first decide what course interests you the most, architecture or a normal B.Sc course.

Yes, you can easily get an NIT for B.Arch because 200+ is a fabulous score.

Here, admission is done through JoSAA counselling based on the rank obtained in JEE Mains. NATA score is also valid in here.
According to the stats of JEE Mains 2017 Paper 2, a rank of 2500 in B.Arch paper helps to get a seat in architecture and planing in VNIT-Nagpur. It is a pen and paper based entrance exam. The fee structure is also feasible here. The fee structure is INR 39100 per semester. Yes, it is affordable as there are many scholarships and loans are also provided for the students who are not capable of paying the fee.
I hope that can you clearly estimates the rank for ONB as well in VNIT through above details.