I think your limiting factor is the lens in good light, and the body in low light. Ideally, one would upgrade both, but which one would depend on how and when you typically shoot. Most of my attempts to get good bird and nature shots are in the middle of the day when lighting is at it's best.... so if it were me, I would upgrade the Lens first.
Besides, a year or so, when you are ready to upgrade the body, there will be newer and better ones out and the current bodies will be at lower prices..... not so likely to happen to the 600mm, particularly after it's recent upgrade.