Combine the apples, sugar, cinnamon, nutmeg and 2 tablespoons water in a medium saucepan. Cook over medium heat until the sugar dissolves and the apples release their juices, 4 to 5 minutes.
In a small bowl, stir the cornstarch into 2 tablespoons water until smooth. Whisk the cornstarch mixture into the simmering juices in the saucepan until smooth. Simmer just until the juices thicken, about 1 minute. Stir in the blackberries and blueberries, return to a simmer, and then remove the pan from the heat. Let cool while you roll the dough.
Preheat the oven to 375 degrees F. Set an oven rack in the bottom position.
On a floured work surface, roll out 1 disc of dough to fit a 9-inch pie plate with about 1 inch of overhang. Roll out the second disc and cut strips or shapes. Pour the filling into the bottom crust. Top with a lattice or cut-out design. Brush the crust with the egg wash.
Bake on a baking sheet (to catch any drips) until the crust is golden brown and the filling is bubbly, about 45 minutes. (Check halfway through; if the top crust is browning too quickly, loosely cover with foil.) Let cool.