Add it to one more unlisted feature that Windows 10 Mobile‘s latest build, Build 10572 brings with itself amongst many other that we have covered here and here. Cortana seems to be available in India now without changing Region, Language and Speech to anything else. In fact Build 10572 brings English (India) as the default speech language in India to all Lumia devices.
Untill Build 10549, the speech language used to be blank in case of all the older generation Windows Phone 8.1 devices and English UK in case of newer devices like Lumia 640 XL, Lumia 540 and others in India. So, it was not even possible to get Cortana on devices like Lumia 525, but now you can see the screenshots for yourself.
Cortana India seems to pack all the settings, but misses some of the interests like Eat & Drink, Events, Getting Around as compared to Cortana in US or UK. Do let us know if you have been able to find Cortana India on your devices.