If anything I suspect the X-Men will be the ones on the shelf longer than FF. X-Men will require a bit longer to build up anticipation given that they have yet to actually end the FOX run of films. Plus there's just so many other characters and teams to figure out. The Mutants are themselves not a single property but almost a franchise built of a bunch of other franchises. Characters with solo potential like Wolverine or Gambit etc. but also multiple associated groups. Not just the X-Men but the likes of X-Force, X-Factor, Excalibur, The New Mutants and Alpha Flight. You have to have a plan in place not only for the mutants as a group of IPs but then also how does all of that fit into the wider MCU and the future mega arcs that they are obviously going to do as a follow up to the Thanos/Infinity Stones Saga.
The FF are MUCH, MUCH simpler, with fewer characters to figure out and decide what to do with in terms of how they can fit into the MCU. And there's less need for a "cooling off" phase due to as already noted, the failures of the FF films. Hugh Jackman's last time as Logan was only two years ago and it was not only well received but it was a grand capstone to Jackman's time in the role. He's pretty much Wolverine in the mind of the public, as much as Patrick Stewart is Xavier and Mckellen is Magneto, all apologies to McAvoy and Fassbender but that's just the truth.
The FF don't have that hanging over their heads. No live action version was embraced by the masses. While Marvel has been good at working on the fly with integrating things that were unexpected but with far reaching implications like the sudden availability of Spider-Man, the X-Men are a much bigger proposition. Just think about the search for talent alone. With the FF it's basically finding a director, the right writer or most likely writers and five pieces of casting: Reed, Sue, Johnny, Ben and Victor (Not that I'm intimating that they should link Victor to the FF's origin the way the Story films did, but I'm saying THOSE are the characters that above all else NEED to be cast in a pitch perfect manner. No one cares if Willie Lumpkin is subpar, but you can't whiff on Doom and Mr. Fantastic again.)
Now think of what figuring out the mutants as a whole involves. It's just THAT much more.
And if I can project onto Feige and Co. for a moment... While it's more work to piece together the plan for the Mutants in the MCU I think that there's more challenge in bringing the Fantastic Four and I mean that it's a more rewarding challenge. This is Marvel's first family. THE FIRST FAMILY. The ground zero of the Marvel Universe (Please nerds... Don't bring up Namor or the Android Torch, you know what I ****ing mean...) and the seeds of everything that was to come after up to and including the mega successful change over from a company all about comic book publishing to one that's now a movie making empire. The X-men have had real success as a film franchise already, up to and including spin offs. One can quibble about the level of quality or how accurate to the source material they have been but there are all told 13 Mutant based films. Again, cry all you want about their quality but that's 12 times post the release of the first X-Men film a studio thought these characters held enough potential to green light a little over a dozen separate films with budgets in the hundreds of millions of dollars.
They are at this point a PROVEN property even if there are fans that don't like the final results of every attempt at bringing them to the big screen or that not every film has been a smash success.
Again, perhaps projecting onto Feige and Co. but... I can see a need, a desire to want to succeed with the FF on the big screen where the other productions failed as a matter of pride. The X-Men will be there now to play with and I am sure that they already are planning what to do but since it's more complex it's gonna take more time, and the fact that the characters have been on the big screen consistently for 19 years already may lessen the innate drive to push out yet more films as soon as possible, share holder pressure notwithstanding. Feige has got more than enough clout at this point to be able to say "Sure, we're working on the Mutants... But I want to get the First Family Of Marvel in order immediately, so while we brain storm what to do with the Mutants in the long run let's focus in the short term on bringing Stan and Jack's original Marvel heroes into the mainstream sooner than later." I can see how that, cracking that nut I mean, being a feather Feige would want in his hat, even with all he's accomplished.
I'm more than prepared to be proven wrong. The X-men are popular and had a place in the pop culture via their films before the MCU was even a thing. There are certainly compelling arguments to be made in hitting the ground running with them. But I just think the relative simplicity of the FF as compared to the Mutants along with how it would be a more satisfying "win" to finally get the FF right on the big screen... Yeah I think we'll be seeing the Four before we see Logan, Scott, Xavier and the rest again.