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.
There’s an argument to be made that an alternate timeline isn’t the same timeline, and therefore, the 24th century temporal prime directive would not apply. It may just be conventional application of the prime directive, as it applies to pre-warp civilisations. They’d likely try to get them back, and either give them the choices of keeping quiet, or having their memories of the future altered/erased, to avoid interfering with that iteration of Earth’s development.
If there is no way of them going back, then they would likely get the standard time-displacement reintegration package. The circumstances are close enough, and it would hardly be the first time ancient humans from over 3 centuries ago would crop up.