This part comes right after the description of the killing of Barzini:

By then the people in Barzini's office might have known about Barzini's death.
So Tessio must have been killed first.
How the hell did the Corleones organize all this without mobile phones?

The novel mentions that several top capos switched to the Corleones immediately after the hits were carried out. Because of this, my thought has always been that the Corleones were able to orchestrate things so well because they already had moles inside the other families. So the Barzini man who received the call might have known Barzini was doomed or dead, without letting on to Tessio.

It's still possible that the news of Barzini's death hadn't reached the people who Tessio spoke to yet.

Tessio's call was to Barzini's office. However, that doesn't mean that Barzini was there. In fact, from the novel we can infer that Barzini's murder and Tessio's call occured simultaneously. Tessio's murder could occur anywhere and at anytime. When and where it occured was just convenient to catch him off guard.

However, as far as the movie goes, it is a wonder that the Corleones were able to track the movements of the family dons in order to murder them apparently within a limited time frame and on the same day. Of course, in the novel only two of them are murdered, so it was simpler.

Yes, it points up the close timing of those assassinations. Had one of the Dons heard about the assassination of the other(s), he'd have gone to ground instantly. Then there'd be another war.

I'm guessing Barz was already dead when Tessio got it.

The more I think about it, the more I believe, it was the other way around. They needed Tessio to do his last phone call. So Barzini could be sure that everything was going right. After that they even could have kept him incomunicado until Barzini and the rest were killed.

I don't believe Michael would have trusted a mole inside the Barzini family. That would have been too dangerous.

The killings went in some synchronicity, but some uncertainties had to be taken into calculation. For example, Neri had to improvise when carrying out the killing of Barzini, because the driver didn't go away.

The Corleones wouldn't be able to plan it minute by minute, because at any time a minor detail could change the time table.

I also think, they had to make sure to kill Barzini before Tattaglia, because Barzini would have been alarmed, if he got a phone call from the Tattaglias. And the Barzini killing plan was more fragile. On the other hand Rocco could jump in at any time to kill Tattaglia.

I'm not so sure if that was improvising. I think they expected the driver to not go away. His boss, Barzini, was more important than a fine that could easily be disposed. Barzini had at that stage probably also police and politicians in his pocket. That the driver waited in the car before the supreme court speaks for itself. They never expect someone like Neri, who used this opportunity to stall time.

In the novel Barzini doesn't come out of a court. Neri waits for him at the Rockefeller Center.

Neri is too early. Then he wants to send the car away. ( I guess because he wants to reduce the risk.)
" “Let me see your license and registration,” Neri said. He had been hoping to get the driver to go around the block but there was no hope for that now. Out of the corner of his eye, Neri saw three short, heavyset men coming down the steps of the Plaza building, coming down toward the street. It was Barzini himself and his two bodyguards, on their way to meet Michael Corleone."
(...)
"He put three bullets in Barzini’s barrel chest before the other three men unfroze enough to dive for cover. By that time Neri had darted into the crowd and around the corner where the car was waiting for him."

It was a highly risky mission.

Good analysis D. Yes, it was risky. So many things could go wrong. How about a simple firearm malfunction or a passing patrol vehicle? What is Neri had muffed the shots or one of the bodyguards had fired back not to mention "pain in the ass inncoent bystander"? Neri knew the plan, but had to be prepared to improvise because he couldn't be sure that his targets would cooperate.