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.