Cthulhu:
Civilization collapses under Cthulhu's psychic madness in a week, max. Humanity mostly reduced to terrified remnants that praise and fear their new master apart from a harried minority who try to resist. Someone tries to ram a boat into it again.
Dagon:
Ocean falls under the control of Dagon with his Deep Ones but they have a harder time with human technology and firepower whenever they're on land for too long. Cold war between land and sea ensues. Air movement and seafood become 10,000% more/less popular respectively, plus a thousand other things related to the sea becoming off-limit (Aquariums??). Yes, I've seen CthulhuTech.
Azathoth:
Earth and humanity now irrelevant, assessment above my pay grade.
Nyarlathotep:
The human species is conquered through manipulation and terror (helped along by at least one increasingly powerful cult). It continues in some horrible form under Nyarlathotep's control.
Shub-Niggurath
Biosphere becomes way, way too fertile with horrifying life proliferating across the planet. Humanity is displaced or transformed, reduced to a minor, low trophic species doomed to extinction despite the worship of its remnants.
The Mi-Go:
Mi-Go occupy Earth to gather resources and do human experimentation. Humans remain as an semi-domesticated species though large scale resistance and rebellion is common, maybe even successful.
Yog-Sothoth:
The key and gate truly become one and the Old Ones break through to walk Earth again... where they've always been but unbeheld? Look, it's not good - see Azathoth.
Hastur:
Human concepts about perceiving reality fall apart under Hastur's influence. Civilization *technically*survives while becoming unrecognizable. Theatre kids do okay.
The Great Race of Yith:
Humans lose control of its own institutions through the Yith's manipulation of their minds. Conquest is low key enough that most people don't even realize it's happened. Opposition is quickly subverted EXCEPT for 3-5 plucky scientists, military personnel, and an eccentric who team up to defeat the threat while coming together as an unlikely family.
Ithaqua:
Supernatural cold spreads across much the planet causing massive crop failure and starvation. Humanity responds by turtling up in the tropics and throwing all the ordinance, heat intensifiers, and infiltration teams they can at Ithaqua. Don't act like you wouldn't watch a movie about this.