addressing only the question asked:
Let's make some simplifying factual assumptions: (1) the ship is a US registered ship on the High Seas; (2) the helicopter shot down was US. And, that all US assets are available to POTUS - just as all the assets stated are available to the "pirates" (who, from the facts, are best classed as transnational violent non-state actors).
The answer is not complex legally.
1. POTUS could rely on his inherent emergency powers to take all necessary military and civilian steps to address the problem; but, because nuks are involved, an emergency closed session of Congress (to generate an AUMF - Authorization to Use Military Force) would seem wiser. That would not take that long and, during that time, force planning and deployment would be ongoing. Then ...
2. The "pirates" (TVNSAs) are clearly a designated "hostile force" (because they shot down the helicopter, are holding hostages and have nuks), as to whom "kill or capture, any time, any place" becomes the ROE.
3. All military options are legally on the table, subject to principles of distinction and proportionality (since there are civilians on board), but more particularly to the practical issues of nuclear fallout. Presumably, this problem would fall into the hands of
SOCOM's JSOC, where we would reach a highly classified stage of what would be done - as opposed to what legally could be done.
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