Every light second away from Earth(speed light travels in a second) would add a 1 second delay.
It takes around 13 minutes for a signal to reach mars and vice versa.
Around about where the moon is on average is one light second.
It would take 8 minutes ish for a signal to reach the sun. If it went out we wouldn’t know for 8 minutes.
The universe could already be over on the other side, someone could have initiated a vacuum decay.
We would know immediately if the sun vanished, because the earth would be flung out of its orbit. The sun’s gravitation, which keeps the planets in their elliptical orbits, is not relative to the speed of light. That’s part of the thought experiment that led Einstein to postulate his theory of space-time.
Assuming we were still alive to observe it (and still in the same orbit — which we decidedly would NOT be), the light from the sun would expire about 8.3 minutes after the sun did.