Nuclear strike orders do not require high bandwidth. The orders contain targeting information and arming codes, all of which is pre-programmed on the missiles (the target package is specified with a "pointer" -- the computers on the missiles already have all the details, the order simply identifies which set of details should be used). For ICBM crews it is conceivable that orders could be transmitted on high-bandwidth systems, but for submarines that is not even possible (ELF bands have to be used because of the physical properties of ocean water).

The basic launch sequence is this: a missile crew receives a series of characters and both missileers write the characters down. They check against a codebook and verify that they both heard the same message and that it is a valid launch order. If the message is valid, they unlock the launch keys (from a box secured by two combination locks, each missileer knows the combination for one lock), insert the keys, enter the missile launch unlock code, enter the warhead arming code, turn the keys, and then that's that.

(There are several other steps -- go watch a video of a test launch and listen to the operators read off the checklist, there are quite a few steps. If you are wondering why such information is public, it is meant to reassure the other side that there is no possibility of unauthorized or unintentional launches -- we want our adversaries to know how we control our nuclear forces so they don't panic and preemptively attack us)