Every nations administrative structures are in urban areas, and they tend be centred in a small area. A bomb will destroy almost all administrative structures in a city, multiply that and you have a nation with no governing structures intact, without the means to deliver or organise anything.
Surely, India's top level government will survive.
In india, the administrative structure is totally worthless and thats the reason why most of the India works in a very ad-hoc and corrupt fashion. While in west, we go for formal judiciary, economy, employment, education, infrastructure, healthcare etc, in India, everything is adhoc. Their majority prefers intervention of elders and kangaroo courts than judiciary. Their majority can only afford quacks than real doctors. Their majority believes in mob policing than proper formal cops -- heck they have very few cops per 1000 people. Their economy is mostly informal and based on cash. Education is also highly privatized in the hands of private schools and government schools are disfunctional. Electricity is unreliable. They still use fuels like wood/cow-dung a LOT.
At worst, they will loose their transport infra and will have to resort to ancient ones -- like they did in their lockdown period.
Over a period of 1 year or so they will be able to get most of the rudimentry administration back.
India is a living chaos. Its pretty hard to kill a living chaos like India. Its pathetic but it survives anything you throw at it.
Then there is damage to the water supplies,
damage to agriculture and the eco-system,
long term cost to the people who survive the nuclear explosions,
long term effects from the radiation cloud and effects of radiation on agricultural productivity.
The disruption to the economic supply system,
None of these things cause instant or even near term death. At worst it makes your life shorter. Instead of typical 60 years of life expectency, people in India will have to make do with 40 years for sometime. Actually, I am not sure even it will go down. Radiation looks a hell lot scary than it really is.
Secondly most of India is still very informal and adhoc in its living and infra. They do not depend upon modern and more centralized infra as much as say americans do. A week of failure of electric system will make USA cities buckle, while in India it will be another summer.