This code adds custom email template "Profile Completeness - Profile is complete" to be used in the "Ultimate Member - Profile Completeness" extension.

um_email_notifications_profile_is_complete.php

<?php

/**
 * Add custom email template "Profile Completeness - Profile is complete".
 *
 * @param  array $emails Emails list.
 * @return array
 */
function um_email_notification_profile_is_complete( $emails ) {

	// New email templates.
	$custom_emails = array(
		'profile_is_complete'	 => array(
			'key'            => 'profile_is_complete',
			'title'          => __( 'Profile Completeness - Profile is complete', 'ultimate-member' ),
			'description'    => __( 'Whether to send an email to an admin when a member\'s profile is complete', 'ultimate-member' ),
			'recipient'      => 'admin',
			'default_active' => true,
			'subject'        => '[{site_name}] Profile is complete',
			'body'           => '{display_name} has just completed a profile on {site_name}. <br />To view their profile click here: {user_profile_link} <br /><br />Old role: {old_role}<br />New role: {new_role}',
		),
	);

	foreach ( $custom_emails as $slug => $custom_email ) {

		// Default settings.
		if ( '' === UM()->options()->get( $slug . '_on' ) ) {
			$email_on = empty( $custom_email['default_active'] ) ? 0 : 1;
			UM()->options()->update( $slug . '_on', $email_on );
		}
		if ( '' === UM()->options()->get( $slug . '_sub' ) ) {
			$email_sub = empty( $custom_email['subject'] ) ? $slug : $custom_email['subject'];
			UM()->options()->update( $slug . '_sub', $email_sub );
		}

		// Template file.
		$located = UM()->mail()->locate_template( $slug );
		if ( ! file_exists( $located ) ) {
			wp_mkdir_p( dirname( $located ) );
			file_put_contents( $located, $custom_email['body'] );
		}

		$emails[ $slug ] = $custom_email;
	}

	return $emails;
}
add_filter( 'um_email_notifications', 'um_email_notification_profile_is_complete' );

/**
 * Send an email to an admin when the member's profile is complete.
 * 
 * @param  array $result
 * @param  array $role_data
 * @return array
 */
function um_email_notification_profile_is_complete_send( $result, $role_data ) {

	if ( (int) $result['progress'] >= (int) $role_data['profilec_pct'] ) {
		$user_id  = get_current_user_id();
		$userdata = get_userdata( $user_id );
		$old_role = implode( ', ', $userdata->roles );
		$new_role = empty( $role_data['profilec_upgrade_role'] ) ? $old_role : $role_data['profilec_upgrade_role'];

		$args = array(
			'tags' => array(
				'{old_role}',
				'{new_role}',
			),
			'tags_replace' => array(
				$old_role,
				$new_role,
			),
		);

		$emails = um_multi_admin_email();
		if ( ! empty( $emails ) ) {
			foreach ( $emails as $email ) {
				UM()->mail()->send( $email, 'profile_is_complete', $args );
			}
		}
	}

	return $result;
}
add_filter( 'um_profile_completeness_get_progress_result', 'um_email_notification_profile_is_complete_send', 20, 2 );

This gist is a part of the article How to add and use custom email templates.

Documentation: https://docs.ultimatemember.com/
Support forum: https://wordpress.org/support/plugin/ultimate-member/