You could use a tool like Xenu to check that all of the urls on your clients website do indeed now redirect to https, or to be extra sure, you should add a redirect to the websites .htaccess (or web.conf on windows hosting) to force a 301 redirect on all http pages to https.
As for indexing http, if they redirect to https correctly when you view them, then you'll just have to wait for google to catch up. No harm done in the mean time.