I'm trying to find out what capabilities of the Nautilus were classified when it was first launched. When was the public made aware that we had created the first nuclear powered vehicle? Was the fact that it could launch nuclear missiles known?
Small point of correction.
But the Nautilus as an SSN was a nuclear powered Attack Sub, and both did not carry nuclear tipped cruise missiles, nor was it built with SLBM tubes.
While the USN did develop a nuclear torpedo it would not enter service for several years after the Nautilus was launched.
Previously there had been many experiments and limited deployment of the Regulus Cruise missile. Developed from the German V1, later versions could carry nuclear warheads at Mach 2, and over 1k miles. The USN actually had a handful of subs that operated on nuclear deterrence patrols using the Regulus. But as you can see they took quite some engineering and design work to include the cumbersome missile. And they could only carry a handful of them at a time. The sub was required to stop, surface, and roll out the missile before it could launch.
One easy way to check on what was "known" is to search newspaper accounts. The fact that Nautilus used a nuclear reactor was known before it was actually completed — that the USN was making an "atom sub" dubbed "Nautilus" was announced in December 1951, a little over two years before it was launched. This was widely reported (e.g. "New Atom Submarine is Called Nautilus," New York Times, 15 December 1951, p. 19).
It's harder to get into fine-grained capabilities this way, but it's worth noting that by 1953 it was reported that the Nautilus would be able to operate fully submerged for long periods of time, unlike diesel submarines, because it could manufacture its own air easily. The fact that the Nautilus had a "blunt bow" and did not need to be as "aerodynamic" in the water was discussed in the popular press, as was the fact that it was designed to move much faster than other subs, and the fact that it was going to be likely much larger than previous subs. The fact that there was considerable shielding for the reactor was mentioned, but no details given on how thick it was, however it does note that the shielding was made up of layers of lead, steel, polyethelene (to absorb neutrons), and then one more layer of steel. Reporters were allowed to see the non-nuclear parts of the ship. ("Blunt-Nosed Atomic Sub Designed for Long, Speedy Undersea Missions," Christian Science Monitor, 21 January 1954.)
So that's some indication of what was and wasn't secret. To get into the real nitty-gritty of it you'd need to know specifically what you were looking for, or get ahold of the classification guide they must have produced for it (it was common by the 1950s to produce these for major experiments and projects — they laid out exactly what level of classification any given statement or information about the project was, because while much might be secret, not everything was).
News about the Polaris missile development broke in late December 1956. ("Navy to Get Own Atomic Sub-Launched Missile: Polaris Weapon to Have 800-Mile Range; Can Be Unleashed From Deep Underwater," Los Angeles Times, 30 December 1956.) Over time the mileage varied (it was upped to 1,500 miles in an account a year later). There was considerable press coverage of the US ballistic missile programs of the 1950s, including but not limited to Polaris. These were very expensive programs (much more expensive than the Manhattan Project when taken together), and so efforts were made to keep the public interested and enthusiastic about them. They also used a substantial amount of human labor, so there was not much hiding of the fact that they were under way, even if many details were kept secret.
