Dude, this trailer does not imply the Weapon X program gave him his healing power, he clearly had that for a while as evidenced by the scenes of Wolverine in the Civil War and in WWII without having aged. You can't do that without a healing factor. Weapon X gave him his Adamantium Skeleton which makes him indestructable.

X-Men Origins: Wolverine
Trailer
(2:24)