With the recent addition of the Alolan Exeggutor form, will Exeggcute evolve to the old form, or the new one (with a different move pool and types)?
It will not be like the main series, because it depends where (in-game) you evolve it. From Bulbapedia (emph mine):
In Alola, Exeggutor has a regional variant that is Grass/Dragon. It evolves from Exeggcute when exposed to a Leaf Stone. All Exeggcute in Alola evolve into this form regardless of their origin.
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.