If it is showing as premium then you should be good to go! Sometimes a quick log out and back in again is needed. If you are still having issues, try a clean reinstallation of the application.
If your account is still showing as free then you have not logged into the account you have upgraded. Double check your details, its amazingly easy to accidentally have two accounts, one with a Spotify username and another using your Facebook details. The username of the account you have upgraded is shown in the email receipt/confirmation. You need to log in using that username unless it is a number in which case you need to log in using your Facebook details.
If you can't find your account or you have subscribed on the incorrect account, don't subscribe again as you might be billed twice, instead get in touch with the customer services team directly using the online contact form and they will be able to lend a hand.
If you get an automated email reply back directing you to the community or help pages, you need to reply directly to that email (even if its from no-reply) and one of the customer services agents will get back to you as soon as possible.
Alternatively, if you prefer support via Twitter, please tweet @SpotifyCares.
I sent an email to the support team as you mentioned and I received the automated message so I replied back to them as you suggested to the No Reply mailbox and I received a reply in blank. My account has still not upgraded to premium and I am getting desperate.