India's first submarine was INS CHAKRA nd first nuclear submarine is INS ARIHANT..
Its "INS Arihant".
ins arihant
The USS Nautilus (SSN-571) was the world's first operational nuclear submarine.
USS Nautilus
India's first indigenously designed nuclear submarine is known as the INS Arihant and is the lead ship of India's future Arihant-class nuclear submarine fleet. The symbolic launch of the vessel was held on July 26, 2009. The name "Arihant" means "Slayer of Enemies" in Sanskrit. It will feature an 83 megawatt pressurized light-water reactor fueled by enriched uranium. It will contain four vertical launch tubes capable of carrying twelve K-14 Sagarika or four K-4 nuclear-capable SLBMs.
That would be the INS Arihant, -The Indian Navy's first nuclear submarine.
nautilus
"Smiling Buddha" was the code name of India's the first nuclear weapon test conducted on May 18, 1974.
The submarine you are thinking about was the USS Nautilus. It was the world's first nuclear powered submarine and commissioned into the US Navy in 1955
Uss nautilus
USS Thresher