Natasha originated as a nickname for
Natalya, which is Russian for
Natalie; no reason not to use it that way, but since
Natasha is established as an independent fn outside Russia, there seems no point in making it a nn - just use it as a full name in its own right, like everyone else.
The only nn I regularly encounter for
Natalie is Nats, which I dislike. I wouldn't use any of them - seem a bit trite - everyone has a birthday, after all! - but at gunpoint I'd go for
Natasha.