Tasha

Tasha is used, although rarely, as a diminutive for both Natalya and Tatyana.

Occasionally used in Russia as a nickname for Tatyana, but nowhere near as often as Tanya.

This form is never used in Russia. In fact, the only short form of Natalia is Natasha.

I agree with the above user in that I've never heard of Tasha used as a nickname for Natalia. But Natasha is not the only possible nickname. I'm Russian, and my friend, whose full name is Natalia, goes by Nata, another nickname for Natalia.