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:
- 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.
- In the future, Leonard would have been at the concert for Luther to have removed his eye. In the present, he was not there.
- 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.
- In the future, he found everyone's bodies. In the present, everyone was transported away.
- (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.