1. the first dig site is Nem. This is never explicitly called out, but can be deduced since Apet was "recent" and Jiese was "not first". It makes sense that Nem would be first for several reasons: it is located within the borders of Danor (so it's easy to get to), and it's within the Dead Magic zone (so it's far less dangerous to explore).
2. the second dig site is Jiese, in northern deserts of Crisillyir. This is the second-closest ziggurat to Danor, but the first one where they'd have to contend with magical defenses. This is the one where they hired Rock Rackus as a contractor, because of his history with planeswalking.
3. the third dig site is Apet, in the High Bayou of southern Risur. It's not entirely clear why this one was third. In my headcanon, they kicked off the Apet and Mavisha efforts at the same time and worked them in parallel, and then things just happened to come together on the Apet side more quickly.