Well, we all know that the first 3 movies take place with Wolverine not having a full memory of his past. The Xmen Origins: Wolverine explains is, obviously, and is supposed to be modern times not long before the 1st Xmen movie. With 1st class starting during WWII and then jumping to the early 60's during the cold war.
The inconsistency that is between 1st Class and Wolverine is that 1st Class puts Xavier in a wheelchair at the end of the movie, where at the end of Wolverine he isn't. In Wolverine, at the end of the movie where all the young mutants are on the island and running out of the buildings, Xavier (played by Patrick Stewart) IS STANDING next to the helicopter and NOT wheelchair bound. 1st class is supposed to be over 30 years prior to Wolverine, so I don't understand how they can make him wheelchair bound in 1st class when he wasn't in Wolverine.