I've just bought Origins through Gog, and whilst it says that Awakening is included, along with all the other add-ons, none of them are showing in the 'Other Campaigns' section, nor is it showing and Downloadable Content.

What's happened? The Ultimate Edition is supposed to come with Awakening, but it hasn't appeared

What happens when you click 'New game' on the menu? For me a window opens with choice of starting either Origins or Awakening. Are you not getting this?

I have the same problem.
Dragon Age Origins is installed via GOG, but when I go to New Game, the Dragon Age intro starts and I don't have the option to choose what I want to play.

Don't go to new game, go to other campaigns

Other campaigns is grayed out :-(

Then I don't know what's wrong.
I only installed the GOG version of the game the other day (stand alone, not Galaxy) and it worked fine.
If you go to where the game is installed (not documents)
...\Dragon Age Origins\Addins\dao_prc_ep_1 and open the manifest.xml (notepad will do)
Does it show that Awakening is enabled? See Pic.

You could ask support, but it might be a while before you get a response.

Edit: does your game version show as 1.5.13263.0
I just remembered that the stand alone version has the 1.5 patch as a separate exe
The 1.5 patch was solely to fix problems with the DLC's
Does Galaxy give you any extra install options?

The string in the dao_prc_ep_1 file looks exactly like in the image file.
I would appreciate support, the game is fun, and I would love to play the expansion to see what happens next :-)

Have you tried running the game as Administrator?

I installed it as an admin, but I use GOG Galaxy to play.
I don't know how to start the game there as an admin.

Try to run the game as an admin without Galaxy. Go to the folder where you installed the game and right click on the game's .exe file (it's called daorigins.exe or something similar, if I recall correctly). Now select Properties, go to the Compatibility tab and check the "Run as administrator" box. Now apply the change and try to run the game.