What would be detected is a ballistic missile launch; while there's no "law" that every ballistic missile is nuclear, it is certainly the rule and there is no way to determine what kind of warhead an ICBM/SLBM carries in-flight.
As soon as the launch is detected, countermeasures are imitated. The launch/boost time window is ~ 5 minutes long and without prior warning the U.S. would have to hope that by chance there's missile destroyer nearby, in order to try to intercept it. During midcourse the missile is almost undetectable, because it doesn't have an exhaust. So once the last booster falls off, the first window for interception closes.
Once the warhead reenters the atmosphere the second, shorter (approx. 2 minutes), time window would open. Terminal interception is very difficult because the warhead enters at terminal velocity with ~ 14000km/h (~8700mph). If the warhead would split up into several reentry warheads (socalled "MIRVs), a proper interception is almost impossible, even for the U.S. (but afaik the North Koreans are far from having that technology).
Here are two videos from the old and discontinued Nike-Sprint interceptor missile, just to give an impression of how damned fast these things are (and the Sprint goes "only" 12000km/h!). In the second video you can see how the missile turns from black to white due to the enormous friction.