Blizzard has revealed that everyone who opted into the Hearthstone: Heroes of Warcraft closed beta will receive their key. The majority of the invites have been sent already, so there is a good possibility the key is already waiting for you in your e-mail.

Blizzard said that this final round of closed beta keys is sent to those people who opted into the test before January 7th, if you opted after that you're out of luck, but open beta will launch soon anyway so you won't have to wait too long.

Blizzard's community manager said:

The majority of the final round of invites to the Hearthstone closed beta test have now been completed - We have now directly flagged most Battle.net accounts that have met both of the following criteria:

-The account was opted into the Hearthstone beta test before January 7th.
-The account does not have a Hearthstone license already attached to it as of January 7th.


If your account was successfully flagged for beta access, your account was also sent an email with instructions on how to download and install the game. If you cannot find this email for any reason, please go to your Battle.net Account Management page - The Hearthstone license should be visible under “Summary” - “Your Game Accounts” if your account was flagged for beta access. If your account has not yet been flagged for the beta test, we are still going through all of the accounts and we will provide another update at a later time.

The Battle.net desktop app is required in order to play Hearthstone, so make sure to install the Battle.net desktop app first. If you run into issues with downloading or installing Hearthstone or with the Battle.net desktop app, please visit our Support site for more assistance.
    
Thanks Gaming Blend