Yes NEET exam will surely be conducted since a national level examination like NEET JEE cannot be cancelled. So yes NTA is about to make an announcement late evening today so stay tuned to the official website. And about the colleges, we can't really say. It depends on GOVERNMENT orders only and as far as Covid 19 doesn't come under control completely, they won't start. Because health of students future candidates come first anyhow.

Best wishes. Thank you.