In the file wp-login.php we can see at line 586 the following:
/** * Fires before the lost password form. * * @since 1.5.1 * @since 5.1.0 Added the `$errors` parameter. * * @param WP_Error $errors A `WP_Error` object containing any errors generated by using invalid * credentials. Note that the error object may not contain any errors. */ do_action( 'lost_password', $errors );
At line 631 we have:
/** * Fires inside the lostpassword form tags, before the hidden fields. * * @since 2.1.0 */ do_action( 'lostpassword_form' );
That means we can leave the form as is (to participate from the updates) and extend it with action hooks.
For example:
add_action( 'lostpassword_form', 'my_function_to_add_logo' ) );
Everything else depending to the layout can be done via css.