The details of the Pokémon inside an Egg are decided when the Egg is laid, so the way it's hatched doesn't matter. Each Pokémon species requires a certain amount of "steps" before it hatches, when incubating it in a party. One hour in level 3 Isle Avue is the equivalent of 250 steps in the overworld, or 500 with Poké Beans.
When Eggs are ready to hatch in Isle Avue, they will display a question mark over them. At that point you can tap them to start the usual hatching cutscene.
If you're only hatching 1-5 eggs, keeping them on your party, even without a Pokémon with Magma Armor or Flame Body, would be much faster if you're walking the entire time. Isle Avue's advantage is how many Eggs it can incubate at once (18), and that it keeps incubating while the game is turned off. You can breed a large amount of Eggs before going to sleep, soak them in thermal water, add some Poké Beans, and they might be ready to hatch when you check on them the next day after coming back from school or work.