A secret age
The Secret Age was from the late 1950s to the late 1960s/early 1970s—a time after the House of Un-American Activities Committee had caused the JSA to disband and when costumed heroes were not trusted and so stayed out of the limelight as much as possible.
The X-Men would, in all likelihood, be contemporaries of Blue Streak (once known as Quicksilver, and now known as Max Mercury), Captain Comet, the Challengers of the Unknown, the Doom Patrol, the Martian Manhunter during his early years on Earth, the Metal Men, and Metamorpho. Possible, though noncanonical heroes of the era would be the original Blue Beetle (Dan Garrett), the possibly apocryphal Green Lantern Donna Parker, one of the Manhunter characters, the second Sandman (Garrett Sanford), Sarge Steel in his P.I. days, and others.
The exploits of normal heroes like the Challengers of the Unknown and the Sea Devils, in combination with the occasional public exploits of the X-Men, the Doom Patrol and the rest (especially things like the Doom Patrol sacrificing themselves to save a small town from destruction) helped thaw public perceptions towards superheroes and paved the way for the Silver Age to begin in the mid-70s, as opposed to the early 80s by DC’s Zero Hour timeline.
Xavier would be the government’s expert on the booming “x-factor” later “meta-gene”) population, while secretly protecting them from the very agencies he serves, and training them in the use of their powers, for reasons known only to himself. For this reason, the X-Men remained more or less a secret superhero group until the late 80s when the Dominator genebomb created a significant number of metahumans.
At that point, Xaiver took his institute and the X-Men public, because there were so many new metahumans in need of training and assistance with their new powers that there’s no way he could have reached and helped them all if everything remained a secret.
Of course most of the superhero organizations in the DCU would be aware of the X-Men as a group and their purpose long before this point, things like the Crisis would tend to throw the X-Men together with other groups (after all, the Monitor could hardly pass on their assistance) so, on that front at least, they haven’t been a secret group since the early to mid 80s.
In the present day, the X-Men and Xaiver’s Academy basically function in a manner similar to the way Justice League Task Force was supposed to be operating at one time, taking in new metahumans and educating them in the use of their powers.
Since the X-Men started out back in the 60s, the first generation (Cyclops, Marvel Girl, Iceman, etc.) are all middle-aged heroes, similar to the Justice Society prior to being aged to their proper ages in Zero Hour. The second generation of X-Men (Wolverine, Colossus, Banshee, etc.) came onto the team in the late 70s or early 80s and so have a good deal of experience behind them.
The New Mutants may or may not still be students at Xaiver’s Academy. Depending on campaign taste, they may now be old enough to be full-fledged X-Men with a new batch of characters waiting to learn from them.
One extraneous pet addition I have wanted to make (but couldn’t fit in Kansas Jim’s sequence, since it’s focused on the X-Men, of course) was to recreate a Silver Age JLA-type team in the ’60s. It’s noteworthy that such a team did sort of exist in DC continuity, but it was forgotten in some timestream mix-up—it was the team led by Will “Triumph” MacIntire.
My claim (and a primary component of this team) is that the Marvel Family work best as Golden Age characters, and thus they should begin their careers in 1940. They can survive to current times, of course, but their primary context is the 1940s—they make much less sense originating from any other period.
The team, then (which wouldn’t be called “The Justice League of America,” I’m sure) would consist of Triumph, Captain Comet, Mary Marvel, J‘Onn J‘Onzz, Green Lantern Donna Parker, Blue Streak, either Namor (for a Marvel touch) or Neptune Perkins. The team would have formed, led by Triumph, against some alien menace, and would have ended with his exile from the timestream and everyone involving the incident. When MacIntire returned to our timeline in the modern era, his fogged mind naturally confused the early JLA with his team.
Meanwhile, in the past, the team might have stayed together for a time (maybe working with Paul “Manhunter” Kirk or some other action hero alive at the time to fill a “Batman” role), handling several of the classic JLA adventures which can’t be fit into modern continuity.