I have seen this question: "How does the amulet wearer control a Shield Guardian, and when does it act?" But it seems to fail to address the question of if casting a stored spell takes up an action, and if so, whose action: the amulet wearer's or the guardian's?
While it's not specified, the action required to cast the stored spell should be the same as the action used to cast the spell normally, and must be taken by the shield guardian.
When commanded to do so by the wearer or when a situation arises that was predefined by the spellcaster, the guardian casts the stored spell with any parameters set by the original caster, requiring no components.
So the shield guardian is not just, for example, releasing the stored spell - they are casting it.
Most spells require a single action to cast, but some spells require a bonus action, a reaction, or much more time to cast.
Not to sound obvious, but the casting time of a spell defines the time required to cast the spell. Since the guardian is casting the spell, they have to spend the appropriate amount of time.