Google's John Mueller said on Twitter that the mobile first index won't have a specific launch date. Instead, Google will move sites over to this index "as we see they're ready." Here is his tweet:
We're planning on switching sites over as we see they're ready. It's unlikely that we'll have a single "launch date". Details still in work.
— John ☆.o(≧▽≦)o.☆ (@JohnMu) June 20, 2017
John hinted at this before, saying they will launch the mobile first index in batches. So I guess Google will come up with some algorithm to detect which sites are ready to move over to the mobile first index and launch in some sort of order.
Again, all of this is up in the air as John said "details still in work."
Forum discussion at Twitter.