I've thought about this before and up with a theoretical way it would work. Everybody's afraid of the mutants, and then you've got these four people who are in an accident in the process of doing something heroic that saves a lot of lives. So they become celebrities and role models, and then it comes out that now they have powers as a result of the accident. So the mutants look at this and decide to make The Fantastic Four their public face because of the heroism, family values, etc. The four of them do the talk show circuit and everything, and some people start to reconsider their anti-mutant beliefs, and others consider The Fantastic Four an exception for whatever reason, but some think no, they're no better than a mutant. Maybe they were heroes when they were human, but now they're pushing their mutant agenda. And then some radical anti-mutant faction decides to discredit and eliminate them, and that's the plot of at least one movie.
Now granted, I don't think that's what Fantastic Four fans would actually want from a movie, but I think it works as a way of adapting the source material to this idea of a shared universe. What I wouldn't like would be just tossing a mutant and a few references into The Fantastic Four and then saying, look, shared universe.
