Hello,
If you have got a seat in any IITs/NITs while studying M tech at some private engineering colleges, then you can easily go for the NITs/IITs. You just have to get a transfer certificate from your current college. Also you need to go through the admission details so that you can know what kinds of documents are required. Also you don't need to pay the second year fees for the college. But for the getting the transfer certificate, you might need to pay some charges.
Hope it helps