Where is Sator between the Vietnam Yacht and Italy?

Question

Here is the timeline of Sator from Tenet as I saw here and what I understood:

• Sator is born
• He digs up the time capsule in Stalsk-12
• He spends time in Vietnam on his yacht

• Kat kills him and disposes of the body

• Italy scene with the boats
• The plutonium heist

Since we know that in Tenet,

things always happen

there's a gap between the end of the Vietnam scene (the diving woman) and the rest of the timeline, Kat even says that he "vanished".

According to the timeline above, Sator lives while future Sator spends time in Vietnam.

• Did the henchmen notice his reapparition somehow?
• Was regular Sator aware of this vanishment after the Vietnam timepoint, Kat should have said "hey we were in Vietnam..."?

Answer 1

Sator didn't really "vanish." This apparent discrepancy in his timeline is explained by the dialogue among the Protagonist, Neil and Kat as they were discussing when and where Sator could be:

Neil: He gets to choose the time and place for the end of the world. What moment? What does he choose?
Protagonist: [To Kat] You told me about a holiday where you let him feel loved.
Kat: Vietnam.
Protagonist: You said he vanished. What day?
Kat: I went ashore with Max and he flew off, but I don't know what day it was.
Neil: It was the 14th. Ten days ago. He was in Ukraine.
Protagonist: At the Kiev opera siege. How do you know about that?
Neil: The point is, he wasn't on his yacht, so that's his window.
Kat: To go back to that golden moment and have it be his last.
Protagonist: Everyone's last.

The Vietnam holiday was earlier mentioned in Kat's dialogue with the Protagonist (in the restaurant) as they were discussing Sator:

Kat: In Vietnam, I tried to love him again. Thought if there was love there, he might give me my son back.
Kat: We sat on that bloody boat watching the sunsets, imitating some earlier time in our lives. He seemed happy, so I asked him. And he made me an offer. He'd let me go if I agreed never to see my son again.
Kat: I expressed myself. I took Max ashore. He called us, contrite. And when we got back, I glimpsed some other woman diving off the boat, and he'd vanished.

Terms I'll be using in this answer:

• Sator-1 - earlier version of Sator in his timeline (went to Ukraine on the 14th, after earlier arguing with Kat on his yacht in Vietnam)
• Sator-2 - the later version of Sator that went back to the yacht in Vietnam (also on the 14th) that planned to commit suicide (but did not get to commit suicide as he is killed by Kat)
• Kat-1 - earlier version of Kat in her timeline (did not know that Sator-2 was killed by her future self in the yacht)
• Kat-2 - later version of Kat that went back in time and killed Sator-2

there's a gap between the end of the Vietnam scene (the diving woman) and the rest of the timeline, Kat even says that he "vanished".

Sator-1 hasn't really "vanished." Sator-1 flew off to Ukraine after earlier arguing with Kat-1 on his yacht in Vietnam. At Ukraine, he was at the Kiev opera siege, as mentioned by Neil. After Sator-1 flew off, Sator-2 arrives at the yacht. Sator-2 chose this time to arrive at the yacht because he knows that Sator-1 was not on the yacht at that time and because that's the "golden moment" in his life. (Neil: "The point is, he [Sator-1] wasn't on his yacht, so that's his [Sator-2's] window.") (Kat: "To go back to that golden moment and have it be his last.")

Did the henchmen notice his reapparition somehow?

There is no indication in the film that any of Sator's henchmen noticed (or were surprised of) Sator's "reappearance." Since Sator-2 was aware of Sator-1's activities at that time, he made preparations to not alarm anyone of his "reappearance."

Was regular Sator aware of this vanishment after the Vietnam timepoint, Kat should have said "hey we were in Vietnam..."?

Sator-1 was in Ukraine and hasn't "vanished." Based on the above dialogues, the timeline on the 14th goes like this:

1. At the yacht in Vietnam, Kat-1 argued with Sator-1. (Kat-1: "I expressed myself.")
2. Kat-1 left the yacht with Max. (Kat-1: "I took Max ashore.")
3. Sator-1 flew off to Ukraine.
4. Kat-2 arrived at the yacht after Sator-1 had left and before Sator-2 had arrived.
5. Sator-2 arrived at the yacht and met with Kat-2 (who he thought was Kat-1).
6. Sator-2 called up his people and asked that Max be brought to the yacht. (Max was with Kat-1, and not Anna as Kat-2 misleadingly claimed to Sator-2.) (Kat-1: "He called us, contrite.")

Sator-2: I'll get Max brought back. We should share the moment with him.
...
Sator-2: Bring my son back to the boat.

7. Kat-2 killed Sator-2 before Kat-1 and Max arrive at the yacht. Kat-1 sees Kat-2 dive off the boat, thinking Kat-2 is just "some other woman." (Kat-1: "And when we got back, I glimpsed some other woman diving off the boat, and he'd vanished.")
8. When Kat-1 arrived at the yacht, there was no Sator(s) to be seen. Kat-1 wondered where Sator "vanished" to. She didn't know that Sator-1 was in Ukraine and Sator-2 had already been killed by Kat-2 and his corpse tied to the boat driven away by Mahir.

At no point did Kat-1 see Sator-2 on the yacht so Kat-1 couldn't have mentioned Sator-2's appearance in Vietnam to Sator-1.

Answer 2

Nobody from the original timeline knows that Sator was killed in Vietnam.

Sator goes back to the yacht because he wants to re-live the last time he and Kat were happy together. During the scene on the yacht he believes that the Kat he is talking to is the original, un-inverted Kat from that timeline. He does not know that it is the Kat from his future that followed him to the past (not until she reveals her scar right before he is killed).

Note that Sato specifically plans the visit to the boat as a last action before his death from cancer. So it is logical to assume that he arranged his trip to the yacht in a way that neither his henchmen, nor his former self will notice that he spent time with Kat on the boat.

The only witness of either Kat or Sato ever being there is the Kat from the original timeline, but she does not realize that the woman she sees jumping from the boat is her future self.

So to answer your question, what was Sato doing after Vietnam, there's two answers, as there's two Satos existing at the same time:

• The Sato from the future is dead, his body is being taken away by Kat with the motorboat. Note that this second Sato never participated in any of the original events, so him being dead is not a problem for continuity here.
• The Sato from the past, who just keeps on living his normal live and does whatever he has done up to the point were he is first seen in the movie, including paying $9 million for the fake Goya painting. This Sato does not know anything about what happened on the boat, because future Sato arranged his trip in a way that he knew his past self would not find out about.