The Deadpool character that appears in the solo Deadpool movie, as far as we have been told, is part of the post-Days of Future Past revised timeline. That movie removed any trace of the Weapon XI character usually identified as "Deadpool" from X-Men Origins: Wolverine. (Note that Stryker only uses the term "Deadpool" once, and it's not clear that he's using it as a name/nickname.)
There's still a problem with the timeline, though, in that much of Origins happened before Wolverine's interference split the the timeline. In other words, Logan still served in the Agent X program with a mutant named Wade Wilson.
However, all indications are that the two people named Wade Wilson are not the same people:
X-Men Origins Wade Wilson was already mutant from the time we met him, during or soon after the Vietnam War. He was a part of the Agent-X / Weapon-X program from the beginning, and as far as we know, never has any kind of life outside of it. His mutation was a very basic type of enhanced speed/skill/agility (primarily swordsmanship.)
Deadpool Wade Wilson was a former special operations soldier (we're never told where, I don't think) living in New York City in the mid 2010's. When we meet him, his mutation has not yet manifested, until he joins the "new" branch of the Weapon-X program and has it forced out of him. Prior to that he appears to have established a life for himself as a mercenary. This character's mutation is an incredibly rapid healing factor.
The only reasonable in-universe explanation for this is that the two characters are just two totally different people who both happened to be named Wade Wilson. (Presumably, if Deadpool ever runs into Wolverine, we will get a fourth-wall-breaking explanation for this mess.)