Emproxima's solution is a good one, but my achievement didn't pop normally during the original story missions.
I did, however, go to the Expeditions mode in the main menu (I did it after I beat the game, but I imagine it works regardless), and started a Chapter Replay of Siberian Wilderness. This time, Lara got up from the camp and the achievement unlocked.
It's really early on in the chapter, and it's worth a shot for you if your achievement was bugged.