The axe is on the ground near one of the bodies around a cart, east of the bridge crossing the southern river from the lake (north west of Viper's camp and Rothman in the troll trap).
Alcoholics tend to stash alcohol nearby...
Just search high and low for Rimmer's Rum (but not low).
If you mindread Christopher inside the camp, you can find the rum hidden on the cliff by the gate with a couple bodies hanging from it. Ladders lead up to the top of the gate, with a drunk and the guy that gives you the 'lost axe' quest.
