I'm not entirely convinced that this film will be literally set in the 1960s, but rather in an else-world that simply looks like that time period.
Here me out: In Doctor Strange: The Multiverse of Madness, Stephen Strange and America Chavez passed through many different portals to universes that ran concurrently with the MCU's 616. If my memory serves correctly, there was at least one dimension that seemed to harken to a black and white retro time period (looked like the 1920s). Another universe seemed to suggest a prehistoric setting (a la X-Men's Savage Land). And there was yet another universe that looked very futuristic, almost Tron-like. But despite the visual aesthetics of those various universes, they all existed parallel to the 616.
So maybe this Fantastic Four exist in another universe that simply has the fashion and stylistic vibe of the 1960s but is modern in terms of technology. The Incredibles captured that same sixties James Bond-esque element too, but it was certainly intended to feel modern day.
So perhaps the events of MoM, the Loki Season 2 episodes, etc., will create a convergence of sorts that brings the FF into the 616 universe.