Hi there,
Long time reader, first time poster here.
I am not so sure they actually retconned anything.
Here is my take on what happened. Warning: It might contain some fanboy thinking.
The dark future in DOFP is not the timeline from the previous movies, but some (random) alternate Version.
The scene after Logan wakes up is. All movies happened the way they were shown.
Things that need to be cleared for that to work:
- Wolverine/ Stryker: Mystique 'saved' Logan from the River (Potomac?). Maybe Mystique gave him to Stryker, maybe she set him free, maybe he escaped. Assuming Logan was on leave from the military and not remembering anything from DOFP, he would try to get back to duty and back to vietnam. This is where Stryker notices him again and remembers his Claw popping from Mystiques first attempt on Trask. Origins and all Movies following that can happen.
- Sentinels: If the Sentinels were active during the past Movies why did we not see them? Surely they would have used them during that whole Alcatraz business in X3. Considering it took a few hours the government would have had enough time to deploy them or would have stationed some there anyway or would have used them to attack the mutant camp or Stryker, with his good relationship to Trask, would have used them on the raid of the Mansion in X2. The Sentinel in the Danger Room session is based of of Charles' memories from the 70s as a theoretical threat.
- Jean / Cyclops / Prof X: This is getting trickier as this would have happened between the end of X3 and Logan's rising at the end of DOFP. The Jean in X3 is actually the Phoenix and the real Jeans' body is in a Cocoon underwater while her mind is in the white hot room we saw in The Wolverine. The Phoenix nested itself in Jeans mind when she was a teenager. Charles, assuming it is a part of her, teaches/helps her to suppress it.
The emergence of the Phoenix was noticed by Mr. Sinister who developed his obsession for it over the next months/years. After a long time of searching for it he discovers faint traces of it near a dam in Canada and discovers the real Jean and revives her hoping she will lead him to the Phoenix. To get a better chance at it he improves her telepathic/telekinetic abilities through genetic manipulation. While not able to escape Sinister, Jean manages to put Charles' body back together and transfers his consciousness from that coma patient back into him.
The emergence of Sinister and his tries to harness the Phoenix is also what gets Charles and Magneto united again and seek out Wolverine at the Airport.
During that whole time Cyclops was in a coma in a Canadian Hospital because a small part of his mind was trapped in the White Hot Room together with Jean. He regains his consciousness sometime after Sinister frees Jean from her cocoon. He is integral in the defeat of Sinister and the rescue of Jean.
Depending on Jean's and Sinister's involvement in X:Apocalypse he might have noticed the Phoenix back then but lost track due to Jean suppressing it.
- Public knowledge of mutants: Without the internet in the 70s news and recordings of the events are confiscated by the government, eyewitnesses bought or assassinated and some made up terrorist organization is blamed for the destruction of the stadium. Meanwhile redirecting the public focus on the Vietnam war. After some years mutants are regarded as urban legends and only conspiracy theorists talk about them.
- Magneto: After being defeated in DOFP Magneto is licking his wounds for some time and is trying to create a mutant nation in secret on some far away island (Genosha) with only limited success, due to the cover up of the government, but manages to reconnect to Mystique and convinces her that the war against humans was only delayed and not stopped. So she joins him again.
- Cure from X3: It was derived from traces in the syringes that Hank used in the 70s. It was only a temporary cure.