I'm another wildlife shooter and as such, I have to take a bird where-ever it is perched, the best I can do is get myself into a better shooting location; i.e. one that gets me closer, or gets better separation, tonally or distance-wise between the subject and background, in short - the options are very limited to get it perfect in camera.
This also simplifies PP, I'm not that "arty", so PP will be limited to improving that which I couldn't at the time, typically this will be differential sharpening, blurring and/or contrast enhancement of subject and background, it will often also mean spending significant time cloning out awkwardly placed branches, or patches of bird poo in shot
Other times, if I was lucky with the capture, it won't need much work, just basic stuff.
Like Joe, I tend to use the Auto WB decided by the camera, if that looks wrong and the shot has white or grey in, I'll sample that, if not I might have to revert to a standard Daylight, Cloudy, Shade, or manually tweak. If I have a series of shots of the same bird in the same place, I'll decide on a CT and re-use those figures, numerically entered, on all subsequent shots to get consistency.
My workflow order is basically similar to Joe's, although I don't use LR or mess with TIFFs, because I use ACR in CS5 these days.
To get back on topic; Knowing when to stop is when it is as good as it can be and how much time I spend on PP depends on the quality of the capture (primarily subject sharpness, often related to how much cropping it needed to get a good composition), plus things like the bird's pose, etc.
Cheers,