Destiny 2 forsaken issue?

Are you Lv30+ and have completed the Red Legion main story? The first quest in the Forsaken mission chain won't unlock until you have done so, it will appear as The Forsaken Reef on the right of the Director map.
It doesn't need to do an additional download, it will have already done it all around the date the content was released (since you'll be potentially be playing with players who do have the new content), all the purchase does is let you access that content.

You probably already downloaded the content, but just needed a license key to enable it. The key isn't very large (a few bytes) so you wouldn't even notice anything getting "downloaded."

From what I've heard the Forsaken content can't be accessed until your character has at least completed the main quest line in Destiny 2. So a brand new character won't be able to skip directly to it.