In reality does the nuke code still work even when the president is a hostage?

Most of the procedures around the President and the nuclear football are classified for security purposes. But there are two things that would make kidnapping the President for the sake of launching nukes mostly impossible.

1. Any order for nukes by the President must be verified and confirmed by multiple people, most importantly the Secretary of Defense. While there is no legal framework in place for the SoD to veto or deny the order, given the circumstances, it's likely that the SoD would deny it, and face any possible repercussions. Because a nuke is not something to take lightly. Any number of Generals would likely do the same if they knew the president was kidnapped or not under a proper state of mind.

2. A kidnapping of the President would likely invoke a temporary Order of Succession event.

   Article II, Section 1 of the United States Constitution:

   In case of the removal of the President from office, or of his death, resignation, or inability to discharge the powers and duties of the said office, the same shall devolve on the Vice President ... until the disability be removed, or a President elected.

   Someone that's kidnapped or under duress cannot be considered to able to discharge the duties of the office of POTUS. As such, the Vice President (or whomever is next in line) would be the Acting President and Commander in Chief. The President would no longer have the legal capability to order the strike and neither would the Secretary of Defense be legally authorized to verify, confirm, and pass along the strike order by the now powerless President.

Most movies make it seem like the football is a single device that the President activates and everything else is automatized. It's not. It's a series of commands passed along, verified, and processed by hand by the military structure. Action movies almost always ignore this. Movies that tend to get it right are Sci-fi. Any number of Outer Limit or Twilight Zone Episodes. X-Men First Class, where the officers on the Ship questioned the strike Order until Xavier stops them. Even GI Joe 2, where the football is used to signal officers on the satellite, or how Cobra took over the president via nano-machine chameleon technology, because replacing the president is the only way to access the nuclear football without making it useless.