In the X-Men Cinematic Universe, Wolverine's origin in the Weapon X program has been erased. It's possible the Weapon X program never happened in the new XCU timeline.
In "X-Men Origins: Wolverine", he and Sabretooth were both part of Stryker's Weapon X program. In the same movie, Wade Wilson was also part of the Weapon X program and was altered by Stryker into a very different "Weapon XI".
Then the "X-Men: Days of Future Past" movie rewrote everything, making this Deadpool movie possible. However, it also rewrote Wolverine's origin in an as-yet-unspecified manner. The last we saw Wolverine in the 1970s, he was rescued from the bottom of the Potomac River, nearly dead, and in the possession of Mystique posing as Stryker. In our future, Wolverine is still at Xavier's school, but it's not specified whether he has his adamantium claws by then or not.
According to IMDb, Stryker is in the upcoming movie "X-Men: Apocalypse", but Logan/Wolverine is not. So it's questionable whether we'll get a definitive answer to what happened to Logan between those two films.
This being the case, it's possible the writers of "Deadpool" decided to play it safe and disentangle Wade from the Weapon X program entirely. That way, they didn't have to worry about contradicting the upcoming "X-Men: Apocalypse" film.