How will Exeggcute evolve?
At the moment in time, there is no clear indication on how to obtain Alolan Exeggutor through evolving. I, as well with users on reddit, evolved Exeggcutes and they evolved into a normal Exeggutors.
Based on the announcement, it seems to be implied that it can only be obtained through catching for now. The initial announcement of Alolan forms states Niantic will be releasing additional news in regards to this, so there are sure to be additional announcements in the future.
I looked through the Pokédex for Pokémon who have evolution variants just to be sure, and the Pokédex shows which Pokémon can evolve into the different forms.
Exeggcute simple doesn't have that, and Alolan Exeggutor is shown as an alternative form of Exeggcutor alone.