Question

How long does it take to level up Pokemon at the Pokemon Nursery?

Answer 1

Pokémon do not level up in the Nursery — they only make spawn. I am unsure if there is another facility akin to Ye Olde Day Care, but from what I've researched there is not.

It's in Paniola Town in Akala Island. Here's the catch though, it's a NURSERY this time. The Pokemon you leave there won't grow any stronger nor learn any new moves, ONLY lay eggs.
Source

Comments

Guessed as much. The alternative is the Poke Pelago's Evelup Isle, sans breeding ability.

Answer 2

sadly in this game your Pokemon cant level up while placed in the nursery