It has been speculated that Wade belongs to neither the "Stewart" nor the "McAvoy" timeline, since he exhibits awareness of Charles Xavier bearing either of the actors' likenesses, placing Wade in a sort of X-Men metaverse. More likely, the references are part of many times that he breaks the fourth wall—even the "sixteenth wall". This bit of information coupled with the fact that he is forgetful (at least regarding his bag full of ammo) and nothing about his background is revealed in Deadpool opens the door for a world of possibilities, particularly concerning the effect of his mutation on his brain, but it at least seems unlikely that this incarnation of Wade ever experienced being transformed into Weapon XI (Weapon 11, the 11th Weapon), who was decapitated at the end of X-Men Origins: Wolverine (2009).
Some possibilities (which may not be mutually exclusive) are as follows.
(1) Somehow or for whatever causation mechanics, in the Deadpool timeline, Wade's parents conceived him (exactly him) in the 1970s, instead of the 1940s like the original timeline. (This is not necessarily unusual in science fiction, since there are a number of stories in which a character travels to a world that experienced an alternate history different from ours by virtue of an event that happened before the same character or a younger character was born, thus ignoring the butterfly effect.)
(2) In any timeline or every timeline, Wade was indeed born in the 1940s or much earlier, but ages very slowly (much like Wolverine) or had served as something of a "winter soldier" (an operative who is placed into cryostasis whenever not on active duty)—skipping forward through time by the implied means or some other means.
(3) Wade is actually "Wade, Jr." or "Wade 2.0", whereas the man who became Weapon XI in the original timeline (or even the Deadpool timeline for that matter) was his biological father who looked just like him or even an individual from whom he was cloned.
(4) X-Men Origins: Wolverine has been retconned once again (for the second or third time, the 2½th time) and in such a way as its events never occurred in any canon timeline, or an "oversight" took place (again).
(5) Deadpool as a whole is merely a tongue-in-cheek fest that happens to have very dark thematic elements, not to be seriously treated as as part of or adjunct with the X-Men movie continuity.