She had to be inverted in order to help her heal, because Sator shot her with an inverted round. Then she had to be reverted to get her moving forward in time again. The protagonist could not do this with the turnstile in Tallin because Sator's people controlled it in the recent past, and going back would have gotten the Tenet people killed (at least Ives seems to believe so). So the protagonist had to revert her at the only other turnstile he knew about, the one in the airport. He and Neil had breached it a week ago with the airplane so they knew they could sneak in and use the turnstile to revert Kat.
The future person goes on moving forward through the timeline, while the past person continues along his path till he catches up. (For example, the past Kat who sees future Kat diving off the yacht at the end will go along the path we've already seen her take in the film, in which she eventually goes back in time, kills Sator, and throws him and herself off the yacht.) There aren't two separate people, just the same person moving through a particular moment twice/simultaneously.