Tessio is told to make a call and tell Barzini to "start on his way." In the previous scene, Barzini is on his way to the meeting with Michael. Are these events happening serially or at the same time from different vantage points in The Godfather?

To respond to the questions after my answer, keep in mind that Michael Corleone's plan is to coordinate all over New York and eliminate his enemies. He cannot bring them together because they would be too suspicious. So, here is how it goes down:


Brazini is shot, then an hour later his assassin calls Michael Coreleone and informs him that Brazini is dead. Hagen is then told to tell Tessio to phone Brazini (even though Brazini has already been shot and killed). But Tessio thinks he is alive because the plan is for Brazini to meet Tessio and Michael and work out new business arrangements. Of course, Michael keeps this plan so that Tessio will not suspect anything. After Tessio hangs up, he says that he hopes Mike (Corleone) will give Brazini and him a good deal. At this point, then, Tessio suspects nothing because he thinks Brazini is coming. Instead, however, he is met by three hitmen and taken away. Before he goes, Tessio asks Hagen if he can get him off, but Tom says he cannot. Tessio then says to tell Michael "It was just business. I always liked Mike."
 
To answer your questions:1. About Tessio: he would have no way of knowing that Brazini has been hit because that murder occurs in another part of the city and no one is going to call him at the Corleone house, for sure. 2. Hagen never kills anyone. Moreover, no one will kill Tessio at the Coreleone house where there are women present or the Corleones could easily be implicated. Instead, then, they take him away so that the body can more easily and anonymously be disposed of.
 
Also, Neri knew that Brazini would be by the cathedral because previous arrangements had all been made by Michael Corleone. (Neri works for Michael as a hitman.) It makes sense for Corleone to hit his enemies one at a time in different places at different times. Otherwise, the men would be too suspicious. Also, the police may not connect all the crimes to Corleone if they occur all over NYC, thinking different dons are ordering hits rather than just Michael.

In Mario Puzo's novel The Godfather, the killing of Barzini and the killing of Tessio are not simultaneous; they are sequential. 
When Tessio is told "Michael is ready for you now," Neri, who has posed as a policeman in order to give Barzini's driver a ticket so that he could be standing right outside of the car, has already shot and killed Don Brazini outside of St. Patrick's Cathedral, where the baptism has taken place. After Neri kills Brazini, he darts through the crowd and jumps into a car around the corner. In the car he changes out of the uniform and puts on an overcoat. The car soon stops and Neri hops into another vehicle while the first car with the uniform and .38 pistol will be disposed of.

An hour later he is in the mall on Long Beach and talking to Michael Corleone. (Book 9, chapter 31)

The next scene involves Tessio, who is waiting in the kitchen of the old Don's house. Tom Hagen comes into this kitchen and tells Tessio, "Mike is ready for you now." This statement indicates that Michael has received the phone call from Neri, who has made it to Long Beach. So this action takes place at least an hour later.
Hagen adds, "You better make your call to Barzini and tell him to start on his way" only to make Tessio believe that all is going as he has planned so that he will not suspect anything.

In the film, to the best of my recollection, Tom Hagen is not at the baptism. He is depicted in the very immediate aftermath of the baptism/slaughter at the Corleone estate where he informs Tessio that he and Michael are aware of his, Tessio's, betrayal. It can, therefore, be inferred that Tom was not at the baptism, as the sequence of events would not allow for him to be at both places at once. It is then that Tessio inquires of Tom as to whether Tom can talk Michael out of having Tessio killed for his betrayal or, as Tessio puts it, "for old time's sake."
My recollection of the novel is that the series of events that comprise the film's climactic sequence similarly occur somewhat simultaneously, but not completely.
I hope that this helps. If you have the time, the screenplay to The Godfather is available at the following site.

Great answer. So just to be clear - Tessio calls Barzini, Barzini sets out for the meeting, Neri shoots him, the Tessio killed. Correct?
Is that order the same in the novel (just not during the baptism)?
 
Thank you!
 
 

The question does not specify whether Mario Puzo's novel The Godfather or the film based upon that novel with a screenplay co-written by Puzo is the source from which the answer is to be drawn. For purposes of discussion, the following answer presumes the latter.
The scenes at issue occur concurrently in the film. Francis Ford Coppola's vision for this series of scenes as spelled out in available drafts of the screenplay and in the final product clearly depict simultaneous developments occurring at the Corleone family estate, at the church at which Michael Corleone is standing as godfather to his sister Connie's baby, and at the various locations around New York City and Las Vegas where the murder of Moe Green takes place. Coppola structured this sequence deliberately for maximum dramatic effect to emphasize the contrast between the solemnity of the religious service against the systematic slaughter of Michael's enemies among the Five Families of New York's Mafia, as well as the murder of Moe Green in distant Las Vegas, the latter illuminating the reach of the Corleone family's financial interests and abilities to protect or advance those interests.
The scenes with Tessio bookend this climactic sequence of events to emphasize the fate that befalls one of the late Vito Corleone's most trusted lieutenants, the caporegime who betrays Vito's successor, the young, newly established godfather, Michael Corleone.