Will India’s Name Go the Way of Turkey?

Govt minister has rejected this as a rumour [1]

It will be a domain nightmare - both .br and .bh are taken, but I guess .bharat is now a possibility?

The .bharat Internationalized TLD is already available in 22 Indian languages [1]. It isn't available in English however.

I personally don't think .in is going away anytime soon. The question you pose is already being pondered upon, like here [2].