You're only traveling back in time while inverted. They spend most of the main part of the movie not inverted. When they're in Mumbai, or stealing the item from the armored car, Neil is not inverted, otherwise he would seem to be moving backwards, and he would need to carry oxygen.
The only time Neil is inverted, until they both go through the first turnstile to save Kat's life, is when Neil (we find out later) saved the protagonist at the opera house early on, by shooting one of Sator's men with an inverted bullet, and he walks away backwards so I suspect he was inverted, and not just the bullet.
If Neil is Kat's son, then at some point in the future, Neil must have spent decades inverted in order to come back to this time. That's easy to ignore, but it's another big problem with the movie, because it's difficult to stay inverted - you have to carry or make inverted air, and maybe inverted food (they never covered that, but it's a logical assumption if you can't metabolize inverted oxygen). But it doesn't really add up, and I'm surprised no one talks about it, unless Neil is not Kat's son. That makes it more tenable. If the protagonist meets Neil as an adult in the next year, then spends a few years training and recruiting him (and maybe Neil already has his PhD?) then it's not a big stretch to think that Neil may have accrued a few years worth of inverted travel, while running operations, to wind up at the start of it all. That's why, despite the tempting clues implying Neil is Kat's son, I really don't think he is. It just doesn't make sense. Even if he's 25 years old when he finishes his training, he'd have to spend 15 years inverted to get back to this time, so he's actually be 40 years old (as far as we know you age normally in inverted time).
I actually did watch the movie last night, but dozed off half way through. Will pick it up again today. It does make a lot more sense now, however.
- "The same idiots whining about cancel culture are the people who tried to cancel pretty much every video game, book genre, and RPG when I was a kid." - James S. A. Corey