I did notice the award and can confirm it said 2016, but wasn't able to read it for context. I did not take note of the month on the watch, but if January is correct, the following would seem to make the most sense for the film timeline:
2016
Strange receives award toward the end of the year (likely in November, around the time the film was released)
Film begins, Strange in accident.
November 2016-November 2017
Strange recuperates from accident.
January 2018
Strange arrives at Kamar-Taj
January 2018-Winter 2018
Strange trains as a sorcerer.
Winter 2018
Climax of the film. (It is snowing in New York).
The film does provide some justification for Strange's quick training; he has a photographic memory, translates ancient texts with the aid of the internet, and can read books in his astral form as he sleeps!
The thing is, there will obviously be some crossover between this movie and the third Thor coming out next year, and somehow I doubt that Thor will be set in late 2018/early 2019. Still possible, I suppose, depending on how long that storyline takes. It's also possible to justify the opening of Strange as occurring as early as January 2016, if we assume the award is some kind of grant or prize for the coming year. (Or we may have to ignore it altogether.)
The time manipulation elements of the film are not presented in such a way that allow any compression of the film's timeline. However, I am going to go ahead and assume that the three years or so currently missing between Iron Man and Iron Man 2 are the result of the Ancient One using the Eye of Agamotto to defend the Earth against one of the unnamed mystical threats mentioned in Doctor Strange.
Although, where were the folks at the New York Sanctum when the Hand brought the Black Sky into the city a couple of years before?!
EDIT: I keep seeing people thinking that the service member with the spinal injury must be War Machine; did we ever settle that one way or the other?