Did you set the environment variables in /app/.env file?
I have this at the top of that file:
corresponding to the database settings within Local by Flywheel.
For reference, this is step 2 on the bedrock installation steps: https://github.com/roots/bedrock
Oh, and you shouldn’t have an app/public folder if you are setting up Bedrock — see Clay’s step 2 above in which /app/public is deleted.
The only wp-config.php file should be in /app/web and is a very minimal and bedrock-specific:
* Do not edit this file. Edit the config files found in the config/ dir instead.
* This file is required in the root directory so WordPress can find it.
* WP is hardcoded to look in its own directory or one directory up for wp-config.php.
require_once(dirname(__DIR__) . '/vendor/autoload.php');
require_once(dirname(__DIR__) . '/config/application.php');
require_once(ABSPATH . 'wp-settings.php');
Here’s a screenshot of my ~/Local Sites/bedrock_test/ directory: