Saving multiple checkboxes with WordPress settings api

The field name should be uwcc_settings[uwcc_checkbox_field_1][], i.e. turn the field into an array.

Secondly, when saved as an array, you should use in_array() along with the checked() function. But first, let’s put the uwcc_checkbox_field_1 option’s values into a variable:

$uwcc_checkbox_field_1 = isset( $options['uwcc_checkbox_field_1'] ) ?
  (array) $options['uwcc_checkbox_field_1'] : [];

Now you call checked() like so — the 1, for example in the first line, means the default value (as in <input value="Mastercard" type="checkbox".../>) is Mastercard:

checked( in_array( 'Mastercard', $uwcc_checkbox_field_1 ), 1 )
checked( in_array( 'Visa', $uwcc_checkbox_field_1 ), 1 )
checked( in_array( 'Amex', $uwcc_checkbox_field_1 ), 1 )

The full code, tried & tested working: (re-indented for clarity)

function uwcc_checkbox_field_1_render() {

    $options = get_option( 'uwcc_settings', [] );

    $uwcc_checkbox_field_1 = isset( $options['uwcc_checkbox_field_1'] )
        ? (array) $options['uwcc_checkbox_field_1'] : [];
    ?>
    <input type="checkbox" name="uwcc_settings[uwcc_checkbox_field_1][]" <?php checked( in_array( 'Mastercard', $uwcc_checkbox_field_1 ), 1 ); ?> value="Mastercard">
        <label>Mastercard</label>
    <input type="checkbox" name="uwcc_settings[uwcc_checkbox_field_1][]" <?php checked( in_array( 'Visa', $uwcc_checkbox_field_1 ), 1 ); ?> value="Visa">
       <label>Visa</label>
    <input type="checkbox" name="uwcc_settings[uwcc_checkbox_field_1][]" <?php checked( in_array( 'Amex', $uwcc_checkbox_field_1 ), 1 ); ?> value="Amex">
       <label>Amex</label>
    <?php

}

And don’t forget to quote the checkbox values; e.g. 'Mastercard' and not Mastercard. See below, assuming that the Mastercard is not a PHP constant:

checked( $options['uwcc_checkbox_field_1'], Mastercard )   // bad
checked( $options['uwcc_checkbox_field_1'], 'Mastercard' ) // good

checked( in_array( Mastercard, $uwcc_checkbox_field_1 ), 1 )   // bad
checked( in_array( 'Mastercard', $uwcc_checkbox_field_1 ), 1 ) // good