Through the X-Men film universe (X-Men 1-3, X-Men: First Class, X-Men: Days of Future Past, X-Men Origins: Wolverine, and The Wolverine), we see at least three distinct timelines/alternate realities.
The "Original" Timeline
Through X-Men 1-3 mutants are a largely unknown factor. X-Men 1 opens up in "the near future" (relative to the year 2000, when the film was released) with the public largely unfamiliar with mutants or their abilities. The "Mutant Registration Act" aims to identify and register mutants. The following films deal with mutants still being very much hidden, but there is no indication that they are being hunted or even identified in any large scale way until X-Men: The Last Stand when mutants voluntarily come forth. In these movies, the only mention of anything even remotely close to a Sentinel is a robot head in the Danger Room in X-Men: The Last Stand. In this timeline, when mutants appear, police are called, not Sentinels, and
Jean Grey and Cyclops both die (Jean dies twice).
The Days Of Future Past Timeline
In this timeline, Sentinels are used to hunt mutants and anyone with mutant genes.
The whole premise of the movie/Wolverine's mission is that
the assassination of Bolivar Trask in Paris in 1973 sets off anti-mutant hysteria, leading the US government to activate Trask's Sentinel project. In addition to the Sentinels, Trask has a device which can detect mutants (he has it with him at the Paris meeting which he is to be assassinated in). When the assassination fails, Trask already has a number of Sentinels built and ready to deploy, proving that while the updated Sentinels with adaptation ability are still a ways off, the technology for the Gen-1 Sentinels is largely ready - in 1973.
These two timelines are definitively different as
there are no Sentinels hunting mutants, nor is there any massive anti-mutant hysteria in X-Men 1 (at least, not to the level described in Days of Future Past)
The Altered Days of Future Past Timeline
Prior to Wolverine embarking on his mission, Kitty explains that should he succeed,
he will wake up back in their present (2023, I believe), but with history changed around him. Only he would remember the original Days of Future Past timeline with mutants having been hunted to near extinction.
In the end, we see this timeline
with Jean and Cyclops still alive and well in the 2023 present time.
Phew... Got all that?
The present story of The Wolverine opens with Wolverine having nightmares of when
he was forced to kill Jean Grey.
By itself, this doesn't mean that The Wolverine takes place in the "original" timeline, but we see a world where, as with the "original" timeline, there is no indication of mutants being hunted. Nor are there any mentions of Sentinels.
Conclusion: It is reasonable to assume that The Wolverine takes place in the same timeline/continuity as X-Men 1-3.
Right until we get to the post-credit scene
The Wolverine ends with
Logan flying off with Yukio, but only has bone claws.
The post-credit scene takes place later (2 years, IIRC). In this,
Wolverine sets off metal detectors, and is then approached by Professor Xavier and Magneto, hoping to recruit him for an upcoming battle. In the background, we see a commercial for Trask Industries playing on a TV.
The insinuation, of course, is that this scene is meant as a direct lead-in to X-Men: Days of Future Past, and that
Xavier and Magneto are recruiting him to fight the forthcoming Sentinels.
If we assume that The Wolverine and the post-credit scene are in the "original" timeline, then the post-credit scene is not a true lead-in.
If we assume that the post-credit scene is in the Days of Future Past timeline, then we are left to explain why no mutants have are being hunted by Sentinels already.
So... Which timeline does this post-credit scene take place in, or why is the history described in DOFP not reflected in X-Men 1-3?