Also, if the Warg Den has been built, you should see Wargs in your resource bar. They won't spawn on the map, nor does the Warg Den show how much longer until a 'full' warg is generated, as far as I know.
Furthermore, without mods, the minimum number of horses or wargs required for a trained unit is three, so even after you complete the Warg Den you'll need to wait at least 15 turns before you can train your first unit of warg-riders, unless you have other Warg Dens or purchase Wargs from AI factions.