Learn Russian: Пускай новый король подпишет этот указ.

Both are particles used to form a third-person command, meaning 'let' or 'have someone do something'. Пускай is simply a slightly more conversational or colloquial version of пусть. You could easily use пусть here instead, and the meaning would not change at all.

If this is a command, why does подпишет look like it is in the future tense?

Because it is! When you form a third-person imperative with пускай or пусть, you must use a verb in the present or future tense. Here, we want the king to successfully complete the action of signing the decree, so we use the perfective verb подписать. Perfective verbs do not have a present tense, so its conjugated form подпишет is future tense, even though the command is for right now.

Why does этот указ look like it is in the nominative case if it is the thing being signed?

It is actually in the accusative case because it is the direct object of the verb. However, указ is an inanimate masculine noun. In Russian, the accusative form of inanimate masculine nouns (and matching modifiers like этот) is completely identical to their nominative form.

Is подпишет related to the verb писать (to write)?