According to what I read from the KC unofficial wikia, she survived one bomb but she partially survived the other. She sprung a leak, the crew (not sure on this; you would have to be crazy to stay around a nuclear bomb) couldn't patch her up due to radiation levels being too high, and then she started descending into the depths.
the crew (not sure on this; you would have to be crazy to stay around a nuclear bomb)
Experiment's crew, not ship crew. They're like the stage staff who mop the floor and fix decoration for the next show then leave the place after the job is done.
Nagato is already in patches when she arrived, worse condition than other ships.
But many larger ships survived the test anyway, although the pigs and goats inside aren't so lucky.