In X-Men Origins, Logan and Victor Creed (Sabretooth) are shown fighting in the Vietnam War before being sent to execution.
In Days of Future Past, politicians and generals are seen to be signing the Paris Peace Accords, ending America's war with North Vietnam. This means that Logan and Victor's war service is before the moment Mystique attacks Trask at the Paris Peace Accords, and the interruption by Logan, Hank, Xavier and Erik, and therefore does not get affected.
Therefore, this means that Logan and Victor surviving the firing squad and joining Team X with Stryker is before future Logan changes anything in the timeline, meaning the timeline continued with Logan and Victor joining Team X.
So, technically, the way it works out, Logan and Victor fight in the Vietnam War before Victor kills one of the officers. They survive the firing squad and join Stryker's Team X and do work there. One morning, future Logan wakes up in present Logan's body and does the stuff in Days of Future Past. And then, it changes. Mystique, pretending to be Stryker, recovers Logan's body, but Mystique slips up and the real Stryker manages to get the body. He then experiments on Logan, but Wolverine escapes as in Apocalypse.
But there is still the problem with the two Wade Wilson/Deadpools. Unless there are two Wade Wilsons that coincidentally both talk a lot (well, the first had his mouth sewn up, but before that) and both decide to call themselves Deadpool and both get powers, I got no idea on that.
So, simply, yes, X-Men Origins: Wolverine should be canon, but no, X-Men: Origins is not canon.
In even more simpler terms,