Why didn't the algorithm detonate when Kat killed Sator?

Sator has a dead man's switch which will detonate the algorithm and reverse entropy, ending the world.

Sator's dead man's switch by itself will not detonate the algorithm. It just sends out the location of the dead drop (the buried capsule). The people in the future (posterity) are expecting Sator to send them the location of the assembled algorithm (inside the capsule / dead drop), so they could get it in the future and trigger it then.

Since the Tenet team was able to lift the algorithm from the dead drop location, even if the dead drop location has been sent to posterity, it can not be triggered (because it's not there anymore).

Related dialogue (emphasis mine):

Protagonist: [Talking about Sator] He can communicate with the future?
Priya: We all do, don't we? E-mails, credit cards, texts. Anything that goes into the record speaks directly to the future.

Neil: How?
Protagonist: Dead drops.
Protagonist: He buries his time capsule, transmits the location, then digs it up to collect the inverted materials they sent.
Neil: Seemingly instantaneous.

Priya: Sator's lifelong mission, financed and guided by the future, has been to find and reassemble the algorithm.

The people in the future financed and guided Sator to find and reassemble the algorithm, not to trigger or activate it.

Protagonist: The fitness tracker he wears.
...
Neil: It'll be linked to a switch, probably a simple e-mail burst that reveals the location of the dead drop, set to fire if his heart stops.

The fitness tracker's dead man's switch only reveals the location of the dead drop to posterity. If Sator was meant to trigger or activate the algorithm himself, in his present, there would be no need for him to reveal the location of the dead drop to posterity.

Protagonist: [To Kat] You're not there to kill him, you're the backstop.
Protagonist: If we don't lift that algorithm and he kills himself, he takes us all with him.

Note the words: "backstop" and "and". Kat preventing Sator from killing himself is just the "backstop" and not the main part of the operation. The main part of the operation is the temporal pincer to lift the algorithm from the dead drop location. Kat failed to prevent Sator from dying (and actually caused his death herself), but the algorithm didn't trigger or activate. If Sator's dead man's switch was meant to trigger the algorithm, then it's "end of play"—to use Neil's words—as soon as Sator got killed. The Tenet team would not be able to lift the algorithm from the dead drop location, because the algorithm was still inside the cavern / dead drop location when Kat killed Sator.

Protagonist: What's more fanatical than trying to destroy the world?
Sator: I'm not, I'm creating a new one.
Sator: Somewhere, sometime, a man in a crystalline tower throws a switch and armageddon is both triggered and avoided.

Sator says that he is not destroying the world, but "creating a new one" (for the people in the future). Sator did not refer to himself or his present when talking about who will trigger or when armageddon (the algorithm) is triggered.

Sator: Now time itself switches direction. The same sunshine we basked in will warm the faces of our descendants' generations to come.
Protagonist: How could they wanna kill us?
Sator: Because their oceans rose and their rivers ran dry. Don't you see? They have no choice but to turn back. We're responsible.
Sator: Knowing this, do you still want me to stop?
Protagonist: Yes. Each generation looks out for its own survival.
Sator: That's exactly what they're doing.
Protagonist: But not you. You're a traitor. Bringing death to all, because you have no life of your own left.
Sator: When I'm done, life continues.

The main goal of the people in the future is not to destroy the world. Their main goal is their survival, to save their own world, whose environment is so degraded that it can barely support life. Sator (who is financed and guided by the future) detonating the algorithm in his present does not make sense if the main goal of the people in the future is to save their own world.

Protagonist: I'd like to say... That you don't have to do this, Kat...
Kat: Worst thing Andrei ever did to me was that offer he made me.
Kat: Let me go if I agreed never to see my son again.
...
Kat: A chance to help save my child. You can't know what that means to a mother.

Mahir: Ives, she's killed him. Ives, do you copy? She's killed him.
Ives: She jumped the gun. She killed him.
...
Protagonist: Kat! You jumped the gun!
Kat: I couldn't do it. I couldn't let him die thinking he'd won. I knew you'd find a way.*
Kat: Wait, you found a way, we're okay, right?
Protagonist: Yeah, found a way*. Be safe.

^{* lift the algorithm from the dead drop location; also, refer to the "backstop" dialogue}
Kat killed Sator knowing that it will just send the location of the algorithm to posterity, and not that it will trigger armageddon. Throughout the film, it is shown that Kat cares very much for her son, Max. It is the driving force for Kat's actions throughout the film. She was deeply hurt when Sator coerced her not to see Max again.

It would be against her character and previous actions in the film if she killed Sator under the assumption that his death will trigger the algorithm, destroying everyone in the present, including her son.

She "knew" that the Protagonist and team would still be able to find a way to lift the algorithm from the dead drop location.

Overall, there is no line of dialogue in the film that suggests that Sator is to be the one that detonates the algorithm. All relevant dialogue and character actions say otherwise. It is not sure if Sator even knows how to trigger or detonate the algorithm.

The "dead man's switch" (Sator's fitness tracker) is not connected to the time bomb in the hypocentre:

NEIL
Not with a dead man’s switch.
KAT
A what?

PROTAGONIST
That fitness tracker he wears...

KAT
He’s obsessive about his health.

NEIL
It’ll be linked to a switch.
Probably a simple email burst,
revealing the location of the dead
drop, set to fire if his heart
stops.

PROTAGONIST
In effect, his death activates the
algorithm. He dies, the world ends
– no one dares kill him.

Key term is in effect; the email has no role in activating the algorithm, it's a signal to the future that allows the reality in which the algorithm is activated to exist. The bomb that seals the shaft is shown, explicitly and repeatedly, to be activated by a timer.

The location of the dead drop is significant because it allows the algorithm to be recovered and activated in the future. From the sat-phone conversation in the hypocentre between Sator and the Protagonist:

PROTAGONIST
(over phone)
What’s more fanatical than trying
to destroy the world?

SATOR
I’m not. I’m creating a new one.
Somewhere, sometime, a man in a
crystalline tower throws a switch
and Armageddon is both triggered
and avoided. Entropy inverts the
same way the magnetic poles have
switched 183 times over the
millennia. Now time itself switches
direction.

Consider the following reality code:

def FutureErasesThePast(algorithm):
    
    #need to implement algorithm logic here before we ship
    thePresent = False

while (reality == exists):
    if (email == sent and algorithm == buried):
        FutureErasesThePast(algorithm)
        

We can assume the email is sent when Kat kills Sator, but because the Protagonist and Ives retrieve the algorithm before the bomb goes off + prevent it (and themselves, with the knowledge of its location..!) from being buried in the hypocentre, one of the logical conditions for the "FutureErasesThePast" function to be called is not fulfilled.

The film makes it all clear that the Future has identified Sator as the man to end the world in his time. This comes again in the grandfather paradox conversation, too.

The future is not the ones who plan to trigger the algorithm, that would be pointless for them. They want revenge, not just suicide.

Sator's tracker will detonate a fuse which will trigger the algorithm and end of the world. The fuse goes off because of Sator's death, but the algorithm is out of its casing and therefore is not triggered. Thus Armageddon is avoided.

The contraption is like any nuclear bomb, there is an initial fuse triggering the nuclear reaction causing the massive explosion.

Sator's tracker is connected to this fuse. When he dies, the fuse goes off. But the algorithm (which is the fuel for the neuclear explosion) has been extracted safely, just in time, so it's not triggered.