Will the chat feature automatically become available after a player turns 16 years old? (Assuming correct birthday information has been given for the Nintendo Account in use.)

If a player is registered as under 16 years old when they first start playing Pokémon UNITE, they will not have access to the chat feature or any other age restricted features even after turning 16.
They will need to link a Pokémon Trainer Club account after turning 16 to adjust parental controls for the Pokémon UNITE save data.
*Age restrictions will vary depending on region of residence.

Access via Settings > Account Settings > Pokémon Trainer Club

then enter Pokémon Trainer Club account log in details.