Well, it wouldn't have anything to do with what Thanos, Hulk or Tony "wished" for with their respective snaps, it's simply as Rocket Racoon said in the beginning of Endgame when they found out where Thanos were due to his second snap:
"When Thanos snapped his fingers, Earth became ground zero for a power surge of ridiculously cosmic proportions, no one has ever seen anything like it, until two days ago".
So it is the cosmic power surge that emits from the Infinity Guantlet, doesn't matter the wish, that activates the X-gene. Which would be the same type of energy that activated the dormant X-genes in Wanda and Pietro when they were experimented on with the Mind Stone.
I would think that will depend on their approach. Like, will there be an explosion of mutant content? Or will it be a slow roll-out? I don't think the multiverse stuff will have that much to do with mutants aside from Wanda being a big part of it, because big picture of this was planned before the Fox acquisition was a done deal.
Now they are just working in some connections to the Fox X-Men stuff to make that fit a little bit better into the Disney+ library, but they will definitely be rebooting the X-Men for the MCU and they had so much planned in the pipeline before they acquired Fox that they are in no rush to do it, so I bet there will be a slow roll-out.