The original timeline they didn't get taken to the future.
Robin warned them about the destruction and they headed to the lighthouse. On the way to the lighthouse Daisy bailed out (the video clip from the past) and the rest continued on.
Planet goes boom.
While in the lighthouse Robin told them the time machine was the solution.
Everybody (except Robin) died before the time machine was ever completed.
Surface folks completed the time machine based on the notes and pulled them into the future.
Their removal from the past created the new timeline in which somehow the time machine still existed. Wibbly Wobbly Timey Wimey.