Move Auschwitz up? Alt history is so much more chic than time travel these days. Can always go time travel and just plain ol' ignore the bullshit irrelevant to the characters. No plot holes to see here, folks, just the blob stuck in a interdimensional tear, pay no attention to the fat man behind the quantum time dickery. Cryogenics accidents are always fun, too. Random encounter with throwaway mutant #652 that happens to catapult people a few decades into the future or leave them stuck in a rock for the better part of a century or somethin'. Lotsa' stuff!
... alternately, from what I can recall of some of the x-men stuff, most of that years of background was explained away by him and xavier just kinda'... not doing much in the interim. Building up certain sorts of foundations and doing stuff in the background, but not really active. Things not really kicking off until they go full ideological split and then starting to ramp up until the main stage starts, somethin' along those lines.
Could have 'em waiting for a certain sort of critical mass in mutant population, not really having the human (so to speak) resources to push for their goals until a particular point.* Mountains of ways you can explain away empty years if you really feel like it, basically. Can either pick whatever's least disruptive to the rest of canon you're attempting to build or go hog wild and leave a billion different background plots hanging in the wind ala Star Wars or somethin'.
* And if you're looking for reasons why that point doesn't conveniently happen until the years you feel like basing your reboot, the X-men universe has like a thousand jackass-ex-machinas you can blame for it, and then either use as a background plot or main/side antagonist, or have hit with a rock and die right when your story starts up. The massive fustercluck that is marvel's material over the years means you got options.