So the 2024 Great Old One Warlock who uses the 3rd level Psychic Spells feature and somehow has this as a Warlock spell can essentially inflict a 20ft radius sphere doing 5d8 Psychic damage? Interesting.

Doesn't spread around corners anymore, so there could be a wedge of clean air within the area, if the sphere is blocked by let's say a 5ft wide pillar in one angle.

Changes since 2014:

• No longer spreads around corners.
   - It is blocked by total cover. Being a Constitution saving throw it's unaffected by 1/2 and 3/4ths cover.
• Now affects creatures the same turn it appears.
   - Makes it easier to benefit from features that grant a bonus to one damage roll of a spell. Makes it less convenient to use during Time Stop.
• Now affects creatures when the cloud moves into their space.
   - Behaves more intuitively for beings that live in a world with equal and opposite reactions (ie, real world people).
• No longer "rolls along the surface of the ground." "The vapors" are no longer "heavier than air, sink to the lowest layer of the land," nor "pour down openings."
   - Meaning Cloudkill can now move up or down so long as the direction is  "away from you" and can even stay suspended in mid air.

Still vague in 2024: (meaning the DM decides)

• Doesn't define what happens when the cloud's movement vector would go through a wall or the floor. Common sense would suggest it stops or gets deflected in a direction that is still as "away from the caster" as possible.
• Doesn't define what happens if a cloud's movement vector would be heading directly to a forked hallway. Actual physics would split the cloud into two clouds each half the size of the original. Practicality with running a game would suggest just picking one side or the other.

Are you sure about it affecting them when it's cast, and it's just not when the spell moves to an area the person is in?

Quote from ibervang >>

Are you sure about it affecting them when it's cast, and it's just not when the spell moves to an area the person is in?

 Pretty sure.

It states: "Each creature in the Sphere makes a Constitution saving throw, taking 5d8 Poison damage on a failed save or half as much damage on a successful one."

It doesn't explicitly say "when the sphere appears" but it also doesn't say "When a creature enters the spell's area for the first time on a turn or starts its turn there" the way that Legacy Cloudkill does, thus it implies this saving throw occurs when the spell is cast, similar to Fireball (which also doesn't explicitly specify when).

Then it goes on to say: "A creature must also make this save when the Sphere moves into its space and when it enters the Sphere or ends its turn there. A creature makes this save only once per turn."

The "must also" further reinforces that the initial saving throw occurs when the spell appears. Otherwise they would have started with that stipulation on when the saving throw occurs.

Do warlocks even have the ability to cast cloudkill? 

Answer no.

A multiclassed one could. 

Quote from Valtyrr >>

A multiclassed one could. 

Yes, but it wouldn't matter for the specific scenario that was suggested, because that "Psychic Spells" feature only applies to Warlock spells.