As The Hampster King correctly notes, flight time for a ballistic missile launched from Russia to the continental United States (CONUS) is approximately 30 minutes. Although it would be possible for a ballistic missile submarine (‘boomer’ in Navy parlance) to approach closer and fire an SLBM on a supressed trajectory with a shorter flight time a la the hypothetical attack profile suggested in The Hunt For Red October in reality Soviet nuclear submarines (and to the extent that they still operate, those from the Russian Federation) typically patrol in the Barrents Sea and fire over the Arctic, so the flight time is not much less than an ICBM-based launch.
Also, one of the primary functions of SLBMs is to provde ‘second strike’ capability; that is, to be able to assess an opponent’s remaining strategic assets and warmaking capability after an initial exchange and target those areas specficially. Up to the late 'Seventies, US SLBMs lacked sufficient accuracy to target specific installations and were intended to be targetted at ‘soft’ targets (e.g. population centers and manufacturing/agricultural zones) as a deterrent against further attack. Accuracy of Soviet weapons, while not as bad as often suggested, lagged US capability and likely also targetted ‘soft’ targets.
Launches against the US and allies are detected by Defense Support Program Satellites (DSPS) which orbit the globe looking for infrared signatures which are characteristic of large ballistic missile plumes and nuclear explosions, both of which are quite distinct from other natural or man-made phenomena. The DSP satellites and the discrimination software behind it should be able to identify a launch almost instaneously, and it would only remain for NORAD personnel to confirm that the launch is valid (e.g. that the trajectory of the plume is following a ballistic path that would overfly the United States or allies) to issue a launch warning at the executive level.
How fast that would actually be converted into public notice is unknown and essentially untested, especially in the post-NATO/Warsaw Pact era. The policy during the Cold War was “Launch On Warning”; that is, with clear signs of an attack the default policy was for the National Command Authority (NCA, e.g. the President or his successors if he is out of contact or incapacitated) to issue launch orders against a set of plan options, which offers the opportunity for a war borne out of error or misinterpretation. The current response plan (OPLAN)–the details of which are classified–is supposed to specify more selective options, but that just means more decisions to be made. It seems likely that it would take at least ten or fifteen minutes before confirmation of attack and launch orders–which would likely include public notification–to be issued.
Note this assumes a strategic mass attack using large ballistic missiles (whether land- or sea-based). This is the ‘doomday scenario’ that US deterrence efforts were designed to counter but do not encompass many other potential modes of nuclear attack, including fractional orbital bombardment (FOBS, a delivery vehicle launched into a non-ballistic orbit so that it looks like a satellite, and approaching from an unexpected vector), high altitude electromagnetic pulse (HEMP, intended for wide scale disrutption fo communications and industry), delivery via low altitude, low observable bombers or cruise missiles, and of course smuggling a nuclear weapon via container ship or other means; all of these scenarios, while not the same class of strategic threat, would offer little or no warning, and it may not be possible to determine the source for a retaliatory response.
‘Nuclear winter’ (if it is as significant a phenomena as suggested) and radioactive fallout are secondary effects that would occur after an attack, and little can be done about them other than to shelter against them and stockpile resources. Against the attack itself, the only real defense is to be well away from the affected zone where heat and blast effects will destroy unreinforced structures and unprotected people. The initial radioactivy from normal weapons is negligable because over distances shorter than the blast effects the air is opaque to X-rays and other ionizing radiation. However, ‘enhanced radiation devices’ (i.e. the neutron bomb) which are designed to maximum fast neutron yield over blast effects can activate normally inert materials through fast fission and create highly radioactive effects outside the blast zone. Realistically, if you are anywhere near a target area, there is little that could be done in the short period of potential warning that will materially impact your long-term survival.