How to reduce the Practical Lighting (or motivated lighting) from the "Surgical Light" in this Video

8-bit video only gives you 256 different values for Red, Green, and Blue. There is no usable data in the white parts of the video. Your only option would be to manually rotoscope a pale blue solid with the blend mode set to Multiply over the hands, feather it, and adjust the opacity and color to try to put a little blue tint on the hands. Then do the same for the flesh tones and any other colors you want to recover. You won't be able to add any detail,  and it will take a lot of work for very little, if any, gain.  

As Mylenium said, a little color correction may help some, but you're probably stuck with what you got.

Not much you can do. As far as the camera's digital sensor goes, white is white and there's no magic hidden pixels underneath. At best you could apply some basic color correction to tone down the brightness, but it won't recover any extra details. That's the crux with those 3000 Watts OP lights blasting. Overall the video looks okay, though. This is really a case where a second camera with a lower aperture/ exposure and more closely zoomed in to the operation field would probably be a better option next time.

Mylenium