Can I use Leavanny's "Nurturing" attack to evolve a Pokemon that was just played or evolved that turn?
Yes. Since the evolution is from an effect, it can be evolved that same turn.
Nurturing



x2
NurturingChoose 1 of your Pokémon. Search your deck for a card that evolves from that Pokémon and put it onto that Pokémon. (This counts as evolving that Pokémon.) Shuffle your deck afterward.

X-ScissorFlip a coin. If heads, this attack does 50 more damage.
Can I use Leavanny's "Nurturing" attack to evolve a Pokemon that was just played or evolved that turn?
Yes. Since the evolution is from an effect, it can be evolved that same turn.