get_template_part not working with ajax

get_template_part() includes the PHP file, which will break $resp. You need to use output buffering to capture the output into a variable:

ob_start();
get_template_part( 'templates/update', 'profile' );
$data = ob_get_clean();

$resp = array(
    'success' => true,
    'data'    => $data
);