Near as I can tell, the continuity issues between First Class and X1 are pretty minimal. The only two that really stuck out are, one, that Charles went into the chair at the end of First Class, but we saw him walking in the flashbacks in X3 and in his cameo at the end of Wolverine Origins. And two, X1 says that Magneto helped Charles build Cerebro, and new how to make his helmet to block it later. In First class, Hank builds Cerebro and Magneto steals the helmet from Sebastain Shaw.
But you know... wiggle room. Charles has been in and out of the chair a million times in the comics, no reason the same couldn't be true here. Charles was present when Hank was building Cerebro, and technically, the helmet in X1 is slightly different than Shaw's-- it could have been refined by him after seeing cerebro.
And clearly, as in the comic, the friendship between Magneto and Charles is something that waxes and wanes.
So the way I see it, First Class and X1-X3 can easily be the same universe without that much continuity kerfuffle.
The next film seems to me, will work like this: given that Patrick Stewart is in it, that means Professor X is alive in the future segments of the film. That means it is NOT in continuity with the other films since he died in X3. I know the comics, and the X3 post credits scene both offer up ways he could come back, it would be really clunky to have to take the time to explain that in the film that he is likely NOT the center of.
So the FUTURE of Days of Future Past is an alternate timeline than X1-X3. Given Millar's comment about it being T2esque, I think Sentinels from this future are sent back in time to hunt the X-Men in their infancy, because in the future they are about to overthrow the anti-mutant status quo. In that future Xavier is alive, and who knows what other cameos they may give us.
First Class is a common point of continuity, the the timelines split during the events of Days of Future Past. Kind like this:
This seems like the only way the could do it. What I am curious about, is where do they go from there? They could go one of two directions-- they could have all the characters from First Class stranded in the future at the end, which would allow them to have a modern setting (since studios think people dislike period films) with characters from both sets of films.
Or, stay in the past, while continuing to make films in continuity that take place before X1-X3, possibly bringing older characters back in time with them. In the comics, several characters are from displaced timelines. Cable, Bishop, Rachel Summers/Grey, etc.
Either way, they have the potential to merge casts here.
I agree with everyone else though, the real problems are with the things done in X3. So the first option of moving ahead, modern times, in the alternate timeline would let them use both casts and ignore the events of those films. Or maybe, knowing what they do of the future, Charles and Xavier change up their game plans, and the continuity becomes X1-X3 and future films will fit into that continuity.