I dont think that is possible, in my head canon Wolverine Origins just doesn't exist, and Psylocke could be in her 40 in X-Men 3, as could this new angel not be related to Warren Worthington III from the original Timeline since the mutation isn't the same. In X-Men Apocalypse Angel has some king of "hooks" in his wings. If Wolverine Origins isn't canon Deadpool would also make sense. Wolverine origins isn't in that Timeline picture so just more reasons for it not being canon.

The two different Angels and Psylocks are probably different characters. Like the two Emmas and the two Trasks. In fact, there are also probably two different Moiras.

The two Psylockes could easily be explained by her transferring her mind into a younger body (she doesn't seem like the type that would have any moral issues with that in Apocalypse), or even that the one in X3 isn't actually Psylocke.  Is she ever called that in the film?  Maybe it's Revanche.  Maybe it's just a woman with purple streaks.  Especially given that her power seems to be the ability to teleport through shadows, and nothing psychic at all.

The two Angels are different as well, I believe.  Angel in Apocalypse is never once called Warren, does not seem to be able to heal, and has talons on his wings.  The Angel in X3 is actually never called Angel, only Warren, is able to fully regrow his wings after cutting them off, and has no talons.

Re: Erik in 1986, yeah, he has to have gotten out of prison by this point, so he must be on the run or something.  Or maybe Mystique posed as someone to pardon him after she escaped Trask.

We recently went into all this on this thread and addressed most issues: 

http://xmenmovies.wikia.com/wiki/Thread:27006

Well, if that's true that America still had relationships with the Vietnam War even after the Paris Peace Accords, then every post-WWII events of Origins would also be after Wolverine woke up as a body guard in 1973. That would make so much sense, because it was always a little bit confusing imagining that Stryker met Wolverine before he actually woke up in DoFP during the Paris Peace Accords. Stryker just wouldn't react the way it did in DoFP when he saw Wolverine in the Paris Peace Accords if he just met him before. Stryker would most likely try to talk with him, but he did nothing! What a hell? Now it makes sense that Wolverine was actually in Vietnam after the Paris Peace Accords, if it was in a mission led by America, even after the cease fire.

Besides nothing is explicitly said that they are in Vietnam. Maybe they are in another Southeast Asian country! Who knows?

Except, they were clearly shown fighting in Vietnam. Meaning that it was before the peace accords.

There is no evidence to support Styker not recognizing Logan. But rather the opposite.

Seeing Logan's face caught Styker's attention and made him stop running for his life for a second. That wouldn't happen if Logan was a stranger.

Why would he talk to Logan? He tazed his throat and helped Mystique who was trying to kill Trask and Styker. Of course he ran away instead of talk. Anyone who valued their life and isn't an idiot would.

It's Vietnam. It looks like it and Styker even said that he was in Vietnam.

He stopped for a second because, well, he saw a mutant with three pairs of large bone claws getting out of the knuckles of a men. He doesn't seems to recognize him.

Also I can't remember when did Wolverine said he was in Vietnam. Besides, after the Paris Peace Accords, there was another war, involving the USA, in 1973 and in the Southeast Asia: the Communist insurgency in Thailand. Maybe, Sabretooth and Wolverine were in Thailand instead. It doesn't seems plausible that Stryker met Wolverine before DoFP, because the reaction of Stryker in DoFP when he saw Wolverine doesn't seems to support that he actually found him before the Paris Peace Accords.

Plus, I think that nothing in the film X-Men Origins: Wolverine reveals that they were actually in Vietnam. They don't specify the country. Yes, we know that Stryker was in Vietnam, but this was when he found Mystique, Toad, Havok and others. Later, possibly in Thailand, Wolverine was found by Stryker.

So, in DoFP, when Stryker stopped to see Wolverine, was something like "What a hell does he has on his hands?" or "Holy crap, I want those!". A reference to the fact that Stryker became quickly interested in Wolverine when he saw him for the first time.

Also, we don't actually see Wolverine and Sabretooth FIGHTING in the war. They are just in a helicopter and a uncontrolled Sabretooth was shooting at people and we can just see Wolverine telling him to stop. Maybe it was not supposed to do that.

Plus, we see Sabretooth trying to rape that native woman, kills an officer and Wolverine tries to protect his brother. They are in a camp, not in the battle field.

It really looks like a scenario from Vietnam War, but it doesn't seems that Stryker met Wolverine before he woke up in DoFP, because the reaction that Stryker would do in the revised timeline seems to be radically different to those that he did after in the original timeline. In OG, Wolverine joined Team X and that's it. In RV, Wolverine was just captured by Stryker and we quickly jump to the Weapon X, presumably.

It wasn't his claws that made him stop. It was seeing Logan's face that made him stop, then he looked at the claws because he notices something familiar. And like I said Logan and co were just helping his would-be assassinate Mystique after assaulting him. So there is nothing odd about him seeing a familiar face and choosing to run away instead of talk to them in that context. The last time he saw Logan made it clear that he hated Styker for the monster he was. So they weren't friends but enemies.

Styker already knows what Mutants are and has seen far more impressive things than a guy with claws. A shapeshifter, a guy that can shot energy beams from his hands, a guy that can make people pass out, a guy that can make people vomit, etc. These are all he was going to give to Trask Industries to slaughter. Yet bone claws is something he wants? Why?

You can't even argue it's because of the adamantium bonding. Because if this was before they met adamantium hasn't been discovered yet. They found out about that while Logan was on Team X.

Logan tells him "that's enough", meaning an enemy was defeated and Sabretooth was going overboard. If Sabretooth was just randomly firing at people the other guys on the chopper would have reacted differently and he would have been taken prisoner for a war crime. Then there would have been no need for the attempted rape to set-up their execution.

X2 also mentions that Styker was fighting in Vietnam to Senator Kelly. Never mentions anything about him in Thailand. So I bet if I were look at the behind the scenes features of the film or whatever, they would say Vietnam.

Wolverine in the new timeline volunteered for Weapon X in the new timeline. This was implied/stated in Logan.