For the US military its not illegal to refuse to follow an illegal order. The US Marines have this drilled into them, not sure about the other services.

Sure, but you may have next to no information in that little amount of time to make the determination. For all you would know sitting at the terminal, that Russian/Chinese ICBMs were on their way. There was a Radiolab episode about this recently. It's tough to become one of the officers in this position and are heavily selected for those that would act unquestionably on orders. Any hint of questioning orders/superiors/the system gets you kicked out. This was done so that there would be no question in our rivals minds of our intent to follow through with the MAD doctrine.

I can confirm it's drilled into Army soldiers as well, so I'd bet on the air force and navy being the same way.

It's one thing to pledge to this, yet another to actually be in the silo with the alarms blaring and the the orders coming in with seconds to spare. It's impossible to predict in advance how someone would respond to this situation.

How is an order from the chief of the military ever considered illegal?

It worries me that this is a legitimate question. The chief of the military does not define law.

Yes, but in the army you are told to obey its superiors and to follow the hierarchy. The top of the pyramid is the president. Who will challenge him?

Not exactly true. There is a two man rule to launch. On paper the president has the power, but there are extensive safeguards in place.

Not so. The nodes in the command chain have little if any ability to question the order they've been given. Officers are even selected based on their ability to carry out orders unthinkingly. In the US military, an order from POTUS is the ultimate command. The only part of the chain that matters is at the very top, specifically, the armed individuals who surround the President. They can choose to stage a coup or declare the President medically unfit.

They have little authority to stop the order, which is not the same thing as saying they have no ability. In practice they are human beings and a Stanislav Petrov situation is totally possible.

AFAIK the two man rule is to prevent a rogue soldier from launching, they have no authority to question orders from the president.

They are not allowed to override POTUS.

I also wasn't allowed to smoke pot, while enlisted.

Which is to say, the orders may not be followed by the persons at the switch. I make no claim that they will, or will not, be followed. I merely point out that rules aren't always followed.

Correction: Overriding POTUS is illegal and has consequences that do not include the planet being nuked.

They absolutely have the real authority (ability) to refuse the order, and then the real authority to prevent the nuke from being launched (by eating their key, shooting anybody else that comes into the room, shooting the key/console, etc).

Though as I understand it each missile site has its own control and 2 man system. Even if only 1 out of 10 obeyed their orders and launched it would likely be the end of mankind or at least civilization as we recognize it.