I did some digging. And yes - I can confirm that there was actually a nuclear submarine that appeared in Night Vale. It's an Project 971 Shchuka-B submarine.
Soviet designation Project 971 Shchuka-B (Щука-Б) were a series of nuclear-powered attack submarines (SSNs) first deployed by the Soviet Navy in 1986. If we look past the fact that a submarine has no business whatsoever appearing in the desert, which it does not, then we need to ask ourselves why an armed submarine soviet showed up in Night Vale for a 'friendly visit'.
From the photo, it looks like a class that would carry at least four torpedoes and a few surface missiles.
Based on the photo, I figured out approximately where the submarine was, but it's gone now. I know Jane Foster posted about high radiation levels in Night Vale - and if a submarine like this was [...] well. dismantled inappropriately? It might be the source of those reading? But that's just a guess.