Erik and Charles split up earlier in FC than X1 and certainly 3 indicate, Beast invented Cerebro in FC, whilst Charles notes in X1 that Erik helped build it. there's also something of a suggestion that Magneto's dorky helmet was created to block Xavier's telepathy, whilst FC shows it to be Shaw's anti-telepathy helmet that Erik first stole, then copied.
mutants are also clearly a new phenomenon in X1 (witness the discussion of mutants being on the agenda for the UN summit) compared to them having been around for years in FC in 1962!
I think most of that could be rationalized (and I'm not saying that to be hostile, just to play the game of trying to reconcile things). In the comics, Magneto has gone from enemy to ally several times, so maybe Charles and Erik patch up their differences at some point in the future. (The same rationalization could be offered for Xavier walking in the X3 prologue and X-Men Origins: Wolverine
-- his paralysis has been cured and uncured multiple times in the comics.) The version of Cerebro that Charles said Erik helped him build is a considerably later and more advanced version than the one Hank created in FC. The helmet is tricky, though. Maybe at some point Shaw's helmet is destroyed and it's not until the time of X-Men
that Magneto manages to recreate it?
And I never got the impression from the first film that mutants had only recently been discovered to exist. It was clear that there was already a well-established prejudice. The fact that Congress and the UN were having hearings or investigations on the issue doesn't mean it was brand-new, but that it had recently escalated to the point of being seen as urgent. Perhaps that sense of urgency was stirred up by Senator Kelly as a campaign issue after the "mutant problem" had been simmering for decades. That often happens in politics -- an issue that's been around for a long time is made into a cause celebre
and given new prominence. For instance, communism as a significant political force in world affairs was around since 1917 or thereabouts, but the HUAC/McCarthy hearings were in the '50s.
Then there are the discrepancies that are easy to resolve, like between the different versions of Emma Frost in XMO:W and XM:FC. The XMO:W "Emma" is never named onscreen and never given a last name in the credits; she's only identified as Kayla Silverfox's sister. And her version of diamond transformation is very different than FC's Emma (and much sillier) -- just her skin being covered in what appear to be cut diamonds, rather than her whole body changing into a single living diamond. So it's easy to assume they're just two different women. (Not that I figure anyone would have a problem with removing XMO:W from continuity, but like I said, I'm just playing the game of reconciling films, and at this point I think it's still possible to reconcile XMO:W with the others, even FC.)