Originally Posted by FeedOnATreeFrog
After X3 (and The Wolverine), Mutants rise again, causing humans to turn to the sentinels for protection which lead to a catastrophic global war. Rogue, Iceman, etc all die, etc.
Wolverine (or Kitty Pryde) time travels to change some key event that would prevent all this, but gets sent too far back, to around 1930. He/she lays low (as not to change too much in the timeline) until the time is right to tell Charles Xavier.
As a result of the delicate nature of time-travel, and the resulting butterfly effect, the timeline has been distorted, warped, and dragged back in a sense. Many events are happening differently, and often much sooner than they did in the previous timeline, even births.
This act of time-travelling results in the First Class timeline. (thus the discrepancies regarding Xavier in the chair, emma frost, Beast, Havoc, Emma Frost, etc)
(This is also a cheeky way to account for the actors being different, as genetics are not the exact same with the butterfly effect).
This is why Storm and Cyclops (and Havok, his brother), Emma Frost, etc, are born much earlier.
(it was indeed the young-teen Cyclops and Storm that Xavier saw when using Cerebro for the first time in First Class)
DoFP apparently takes place in 1973. By this time, Cyclops, Jean, and Storm would be in their mid-twenties (new actors), ready to be recruited by the time-traveller to stop the event.
This is the original cast's last movie. We never see the original timeline again (as there's nothing really to see except rubble and dead bodies).
Even though they accomplish the task they believe will prevent an apocalyptic future, they still aren't sure that it won't happen for them, and it's something that looms in the back of Xavier's mind for the rest of the series.