As a demon Lord, Fraz-Urb'Luu has many powers. Specifically he can generate a Simulacrum of anyone in his 'lair' – and they last for about ten seconds, give or take. Specially:
Fraz-Urb'luu chooses one humanoid within the lair and instantly creates a simulacrum of that creature (as if created with the simulacrum spell). This simulacrum obeys Fraz-Urb'luu's commands and is destroyed on the next initiative count 20. – Mordenkainen's Tome of Foes pg. 146.
How much time does Fraz-Urb'luu have with this simulacrum? Let us assume he has some kind of 17th level caster on hand (possibly Petrified or Imprisoned) – and this humanoid happens to be a caster that has Wish in their memory-repitoire (i.e. a wizard or bard with a Magic Secret). Possible time-flow in bullet-point format:
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Fraz-Urb'luu must explain to his newly minted simulacrum-creature what it wants it to do. If it wants that illusion-person to cast Wish, this demon lord must explain / include wording of the spell. Not sure how many move-actions this would be. This lord does have telepathy (one of his 'languages'), which might help.
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The creature must cast the spell, this is probably one move-action.
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The creature must include anything it wants with the spell. This would include the full wording of this Wish as indicated by their master Demon Lord creator just moments ago. This may not count as a second move action (?), but it would still require the (re)-wording of the entire phrase once more. Any extensively worded wish will obviously not fit in this time-frame.
How much time do they have together? What RAW exists around the timing, sum total, including this "next initiative count of 20"?
Question: How much time will the Great Demon Lord Fraz-Urb'luu have with his Lair-based Simulacrum from the moment of this casting & creation of his duplication-creature to the end of that creature's life at the end of the next round?
Reason For Asking: In my campaign world this Demon Lord would like to have the ability to cast Wish (and other spells) in his 'home' or lair via this clever mechanic. He also has 26 intelligence and 24 wisdom – he would be intelligent enough to figure out how to do this. This mechanic would be valuable not just for defending himself from brazen adventurers &/or unhappy demon-lords but is also excellent for entertainment &/or creative purposes. Casting a single limitless Wish every second round would be a very useful tool to have.
Other creatures have a similar mechanic, a DM-option allows this for certain hags:
A powerful sea hag might (?) have the following additional lair actions:
The hag chooses one humanoid within the lair and instantly creates a simulacrum of that creature (as if created with th esimulacrum spell). This hideous simulacrum is formed out of seaweed, slime, half-eaten fish and other garbage, but still generally resembles the creature it is imitating. This simulacrum obeys the hag's commands and is destroyed on initiate count o20 on the next round. – Volo's Guide To Monsters, pg. 60.
Best Answer
Here is the answer to the literal question you asked:
Q: How much time will the Great Demon Lord Fraz-Urb'luu have with his Lair-based Simulacrum from the moment of this casting & creation of his duplication-creature to the end of that creature's life at the end of the next round?
A: The simulacrum lasts for exactly six seconds, from one initiative count 20 to the next initiative count 20. During this time it will get one turn.
Here are some answers to other questions you seem to be thinking about:
Q: What sort of action does it take for a simulacrum to cast wish?
A: It takes one action for the simulacrum to cast wish. Move actions are not a thing in this edition of D&D.
You might wonder how it is possible to ever cast a wish with a request that takes more than six seconds to speak. The rules are silent on this topic; one solution might be to speak your wish before casting the spell, and then cast your spell and say: "I wish for that thing I just said to happen!"
Another solution might be to write the full text of your wish on a scroll, and then cast the spell and say: "I wish for the thing written on this scroll to happen!"
Q: What sort of action does it take to explain to a simulacrum what you want it to do?
A: Talking is free and requires no action, although you're still limited by the time available.
Your question talks about "move-actions" but please note that a "move action" is not a thing in D&D 5e.
Q: How much communication can you do with a simulacrum in one round?
A: We have a whole question about this (here), but basically, you're limited to six seconds of communication per turn. The DM might rule that communication via telepathy is faster than communication via voice, but there's no RAW about this.
As you've pointed out, in practice this is even shorter, because the demon lord might need to give a command and then have the simulacrum repeat the command.
Q: Are there any workarounds that would let the demon lord communicate faster?
A: The demon lord might try giving instructions to the caster before creating the simulacrum. For example, the demon lord might say: "Soon I'm going to create a simulacrum of you, and when I do, I will want it to cast wish to accomplish the following effect: (twenty-second explanation)". The simulacrum, when created, will remember this instruction.
The simulacrum can then cast wish as follows: "I wish for the thing the demon lord just said to come true!" They can say that in six seconds.
The demon lord could also pre-write some appropriate wishes down on a series of scrolls, and then instruct a wizard: "wish for Fraz-Urb'luu's Wish #57!"
Q: Are there any other problems with this?
A: You should be aware of this ruling which is used in D&D Adventurers Guild games:
Of course this doesn't directly apply to your home game unless you want it to.
You should also keep in mind that any "more powerful request" casting of wish can fail or backfire, especially if the DM feels you're abusing the rules. The thing you describe here is a pretty clear abuse of the rules; as DM, you could simply choose to never have it fail or backfire, but your players might feel that you were being unreasonable.