Does greyscale/grayscale mode save battery life? Simple question but searching I can't find much evidence. I use an LED device (not OLED), would that also make a difference?
No, because power is still being consumed to display the image. Had the device been OLED or AMOLED, then it would have saved a bit of power, since each pixel is an individual source being powered, and black pixels would not receive power.

The best bet on saving power with LED is if the GPU was processing images in 8-bit, but it does not. If you turned on grayscale and took a screenshot then turned it back off, the image is still in color.