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While it's reasonably clear that

Vanya

is the root cause of the Apocalypse in both the timeline where Five lived after the event and the timeline where he came back to try to stop it, is the mechanics of the event different each time?

It would seem a missing moon would be something Five should have noticed and would have provided some big rabbit holes to chase dead ends down especially given that Luther spent years living there.

The nature of time in the Umbrella Academy seems to be somewhat malleable with the exception of key "events" that must happen (according to the Commission). The Handler notes that the Commission struggles between keeping the timeline intact and giving the humans free-will. So a time-traveler could go back in time and change events as long as they don't change key events.

Spoilers about the entirety of the final episode so read at your own will

There were actually some changes between what Number 5 witnessed and how events unfolded:

1. In the future, Luther was holding a glass eye. In the present, Leonard lost his eye, but never got a glass one. There was no other character that had a glass eye so Luther could not have grabbed one.


2. In the future, Leonard would have been at the concert for Luther to have removed his eye. In the present, he was not there.


3. In the future, Number 5 did not find himself in the rubble indicating that he was not there. In the present, he was clearly there.


4. In the future, he found everyone's bodies. In the present, everyone was transported away.


5. (I may be remembering this wrong). In the future, Number 5 found everyone at the mansion. In the present, they were at the concert hall.

The important thing was that the apocalypse had to happen and that Vanya had to cause it. That was what the Commission sent Hazel and Cha-cha to ensure. How it happened didn't matter. Number 5 even states:

The apocalypse will always happen and Vanya will always be the cause, unless we take her with us and fix her.

He doesn't mention that Vanya had to blow up or destroy the moon. Just that Vanya had to set off a series of events that causes the apocalypse. So it is possible that if they go back and try again without trying to "fix" her, then instead of destroying the moon, she would cause some other kind of apocalyptic event.

It's more than Luther's time on the moon or his assumptions earlier that it had to do with that.
In the final episode, after the mansion's destruction, Number 5 says,

"When I found it, I assumed this place came down along with everything else. But here we are. The Moon's still shining, the Earth is in one piece, but not the Academy."

From this transcript site (emphasis mine)

The comment from 5 struck me since it was really the first implication I caught that the moon might not be shining. Tied in to Luther's time there and our general lack of knowledge about what or how Hargreeves knew about the apocalypse, I just sort of blew it off until the ending. Practically, it seemed like the moon discussion was something that got cut for time/flow issues. "The Moon's still shining" is an odd enough turn of phrase but close enough to a common one (The sun's still shining) that it's noticeable and, frankly, deliberate.

The specifics of the timing/and or trigger may have been somewhat different however:

Look at Grace's embroidery back in episode 3.
I'll try to find an image to include when I'm off of work, but for now you can see one here:The Umbrella Academy's Ending Was Revealed In Episode 3

I think it must be clear that Sir Reginald Hargreeves

knew more then he was ever able to pass on. His meeting with Klaus was interrupted just as he was about to start spilling important info.

This isn't an answer, precisely, but I would say that the in the television series it is demonstrated that events can change. In the finale, Luther never met anyone with a prosthetic eye to yank out, so even if they had been caught in the apocalypse Five never would have found him clutching the eye in his hand. The timeline has been altered. Also, it's only my opinion, but I think that the destruction in the finale would have been much more complete than the shots of the ruined world we saw Five wandering in during the series. The world was relatively intact in Five's lonely post-apocalyptic world. The events of the finale were worse for the planet than the events Five escaped from.