When exactly does Cthulhu rise permanently from R'lyeh?
We know Cthulhu wakes up in Call of Cthulhu and wreaks havoc among the men who discover him before R'lyeh sinks again beneath the waves.
He will rise permanently when the stars are right--that is, when they're in the same position as when he first sank beneath the waves as recounted here and in slightly more depth in At the Mountains of Madness.
This presents an interesting question though as to when exactly Cthulhu's resurrection will occur. His story explicitly states both that the stars are right and that the sailors accidentally succeeded where the Cult of Cthulhu had failed. Nonetheless, human civilization continues after the story though activity by the Great Old Ones seems to amplify in the succeeding years.
However, in The Shadow out of Time, Lovecraft describes two future human empires. First is Tsan Chan, presumably a Chinese-affiliated emperor, which rises to power about AD 5000. There is an unnamed empire of "dark conquerors of AD 16,000"--presumably related to Egypt given the name of their representative among the Yith, Nug Soth. This seems to indicate humanity has at least another 14,000 years as the dominant species on the planet.
Of course Cthulhu will rise permanently once the stars are right again and wreak havoc among humans before causing our extinction before the beetles take over and eventually give way to the arachnids. He apparently wasn't fully ready back in the 1920s, or humanity likely wouldn't have made it to the 1930s.
Naturally I'm curious to know if Lovecraft gave any indication at all as to specifically when Cthulhu and the other Great Old Ones are going to wipe humanity off the planet in his cosmogeny.
Requiescat in pace, Krystle Papile. I'll always miss you.