ILS isn't RNP. They're two different regimes.
Nor does ILS have waypoints.
RNP specifies a level of accuracy. The aircraft doesn't go into "RNP mode" it is just certified to that level of accuracy or not. So while flying an RNP route it behaves just the same as any other time.
I think you're asking whether, during times when the aircraft needs to stay on track a very accurate track, does it aggressively regain track? Yes, it does. Sometimes too aggresively and it starts oscilations.