Not sure if this a fluke, or a firm fix. I managed to get the "play" button working by:
1. Opening battle.net app
2. Clicking 'account management' to open battle.net in web browser.
3. Changing password, while battle.net app was still open.
Once I did this, the 'play' button worked immediately. Literally, immediately.
FYI, this is the total unfolding of events (US region, living in Australia)
1.~approx Midnight Friday (Syd time) I redeemed beta key - did not appear on my account (either online or in-app)
2. Located manual download link for Hearthstone. Play button would not work.
3. Changed password. Hearthstone appeared in account management, but not Play button in App wouldn't work.
4. Cleared cache, re-installed battle.net app, re-installed hearsthone. No change to play button.
5. Waited approx 44 hours. No change to play button.
6. Used first three points above - Play button worked.
Again, not sure if this was a fluke (I know there has been some stress on authentication servers), but thought I would post anyway.
Good luck to all those waiting!