I think the problem with using
Anushka as a nn in the US is that most US nicknames are short and to the point (
Ed,
Bill,
Amy,
Lena, etc.) because nns are used so one doesn't have to take the time to say the full name (
Edward,
William,
Amelia,
Milena).
Anushka looks more like a full name and I don't think many people in the US would see it as a nn.