You seem to be misunderstanding Kitty's phrase:
... once you wake up whatever you've done will take hold and become history. And for the rest of us, it will be the only history that we know.
Wolverine is just required to "wake up", not "wake up in the future where the sentinels exist" as you mention in your comment below the question.
If he does go back in time and alter history, so that Trask's sentinel project gets shut down in 1973 itself, there would be no sentinels in the future. Hence, in the future, or more precisely, in the year 2023, the X-Men and Magneto would not be taking refuge from sentinel attack in the Chinese temple that they gathered in at the start of the film. Rather, the entire history between 1973 and 2023 would stand altered: there would be no mutant-sentinel wars in these 50 years. As a consequence, when the same day in the year 2023 comes, all these characters would be at a different location altogether. This is the reason why people interpret this movie to split the original X-men timeline, since Wolverine altered the original timeline to create an alternate future.
As is revealed in the last part of the film, that 'different location' is just the X-mansion. The sentinel-free future is the happy future, where everyone is alive and well, and the senior X-men have all become teachers/mentors to the younger mutants in the Xavier school.
As a bonus, let me give you an alternative, unrelated analogous story, which explains the situation more clearly:
- Let's assume a scientist couple Alice and Bob are leading a peaceful life in Chicago, in the year 1973 with their child Chris, and a fourth person, Duncan, wants to kill Bob.
- As Bob goes to work on February 23, 1973 at 8:45 AM, Duncan stabs him in front of his office entrance.
- Duncan flees, doesn't get caught, and despite family's insistence, there's no evidence to establish that Duncan killed Bob. Duncan walks off free.
- On March 1, 1973, a grief-struck Alice resolves to erase these events, and resumes working on a time-machine that she and Bob were working on prior to Bob's death. To keep this a secret, Alice relocates her lab to the outhouse of an uninhabited mansion in the outskirts of the city, and visits the lab all heavily disguised.
- On September 19, 1980, a grown-up Chris avenges Bob's murder, by murdering Duncan. Chris gets caught by the police, and is imprisoned while being under trial.
- On September 22, 1980, after years of effort, Alice is finally successful in making the time-machine. But this time machine won't send back Alice as a person, but rather, her conscience, to a specific date and time, to convey the message.
- Alice sends back her conscience to February 22, 1973, and alerts old Alice about what's going to happen next morning. Old Alice convinces Bob about the fore-warning, they both get Duncan nabbed by the police as Duncan is just about to stab Bob at 8:44 AM.
As a consequence of these developments, the post-1973 timeline changed for both families. Chris never murdered Duncan, and therefore, never went to jail. When September 22, 1980 arrives, Bob is alive, Chris isn't in jail, the family is doing fine in their regular house (unless something else untoward happened), Duncan is imprisoned, and no one is present at the outhouse.
That's just what happened in X-Men DOFP, after Wolverine altered the timeline.
