I think you're technically correct, but those arguing with you (particularly those that have served in the armed forces) might cite this as a counterpoint, it takes a lot for it to make it all the way up the chain -

https://en.wikipedia.org/wiki/Stanislav_Petrov

who cares about launching on 7 minutes when nuke subs can strike anyway at any time even after the attack ?

But who would you give command authority for that? Here in the UK, where we wouldn't even have 7 minutes in the event of an attack from Russia, we rely on exactly that strategy (possibly - the contents of the prime minister's "letters of last resort" to our submarine captains are secret and destroyed unopened when a PM leaves office) - but the flipside is that individual submarine captains have both the authority and the means to launch nuclear weapons without requiring approval from anywhere further up the chain.

as far as i know if deep sea subs cannot contact the headquarters for x amount of time they should assume their country was nuked and are supposed to retaliate.

That doesn't answer the question. Who do you give the command authority for that decision? Who gets to say "we cannot contact the headquarters", which amounts to being able to launch the nuclear weapons? And obviously if you want them to be able to launch under those circumstances you have to give them any "launch codes" or what-have-you beforehand.

In the case of the UK, there is a sealed letter in each Trident sub with instructions from the PM in case the nation's leaders have been terminated.

Yeah, I mentioned the letters of last resort, but while the letter will contain orders, the captain has sole launch authority, and it's difficult to imagine the system could work in a way that avoids that - whatever process you require the captain to follow, they could simply decide to follow that process as though the leadership were uncontactable.

I don't know what the actual procedure is; however like with Science Fiction and 'Self Destruct' codes I would think that a multi-agent auth challenge would be required.

The captain and some fixed number of executive officers. Possibly with redundancy for one or two crew being dead. The auth codes would cause a safe to unlock or message to de-crypt.

Presumably the instructions would be delivered to each authenticating member so that all could receive the orders simultaneously.