The legality that Hytens was referring to is not domestic US law per se — it is about the Geneva Conventions, Laws of War, etc. There is no domestic US law that authorizes or prohibits nuclear weapons use. (The main domestic legislation on nuclear weapons in the is the Atomic Energy Act. It does not say anything about their use; it is mostly about their production.) The regulations about the procedures and methods for authorizing nuclear weapons use are contained in Executive Orders/DOD Directives/Other Things Of This Nature that are not legislative in nature (they are not approved by Congress, etc.).

Is the use of nuclear weapons allowed under international law? The International Court of Justice has said no, but the US is not a party to the ICJ and does not consider its rulings binding at all. The US military has repeatedly emphasized over the years that they do not consider the use of nuclear weapons per se to be a violation of international law. That does not mean that military lawyers might consider any use of nuclear weapons legal — e.g., the deliberate, unprovoked attacking of civilians for no military reason would certainly be judged illegal under the laws of war (proportionality, etc.). But even a very cursory reading of US military history post-WWII shows that there are many legal constructions that allow for the use of overwhelming military force, even in the case of high collateral damages, in the prosecution of a military goal. So it is not an exceptionally meaningful statement (I mean, if Hytens had said it was his duty to carry out even illegal orders, that would be noteworthy, since it would be admission of a willingness to commit a war crime; stating the contrary is literally just stating the most basic of regulations) in my mind, and does not tell you what would actually happen in a realistic scenario.

As for the evolution over time: the US has clearly considered retaliatory use of nuclear weapons to be entirely legal. The question is really whether they considered the international legal aspects of a first-strike attack. I have never seen anything that suggests that this angle was considered in the Truman and Eisenhower administrations, both of which had war plans on the books for pre-emptive attacks. It would be interesting to investigate further. There is some documentation about how these matters were talked about in the Korean and Vietnam wars, for example, but to my recollection none of them centered on legal questions (Nina Tannenwalds The Nuclear Taboo is a good source for these kinds of discussions).