I preordered the game on GOG a few weeks ago, and then bought it today by clicking on the "Download" button at the starting page. The email that I received had no free game link, although it had two codes to redeem.

Same here.

Did you try http://www.gog.com/en/page/my_tw2_info (it should show up near the bottom)?

If not, there are multiple people who have had the same experience and I'd contact support about it, being sure to mention that the link above doesn't work for you (because otherwise they'll just link to it).

Yes, I tried that link, and still there was no free game for me I'm afraid.