Let’s imagine that there is an Earth from an alternate timeline where the planet received alien interference in the late 20th century that makes humanity certain of alien life and warp drive (note: not warp-capable), with the early 21st century on technological par with the prime timeline’s 21st century. In this early 21st century, a person accidentally make both a universe and temporal crossing into Earth in the prime timeline and the late 24th century.
Starfleet quickly locates this highly confused person. How would Starfleet handle the situation while abiding by the (Temporal) Prime Directive?
Personally, I would think given the exceptional circumstances, that Starfleet might given the person limited mobility on Earth and only Earth, and eventually allow them to live permanently in the prime timeline if they have exhausted the list of possible ways to return the person to their timeline. This is especially considering that the alternate Earth has already been interfered with, and Starfleet has no way to know the natural development of neither the alternate timeline nor its native earth.
I’m not sure to what degree the PD would apply in a situation like this:
As is tradition, I must point out that whether a species is warp-capable influences whether Starfleet can make first contact, but even if they can, the PD itself applies to all civilizations, and Starfleet is compelled to minimize their interference.
Cultural contamination/interference is minimal, since it’s one individual entering the Federation’s sphere of influence. Ethically, they should probably help the person return home, assuming they want to.
The closest analogue I can think of is Gillian Taylor in Star Trek IV, and the Federation put her to work in pretty short order.