Is it possible to build a 1080p OSD webcam shield? What do I need to build a shield capable of receiving 1080p video from USB camera, timestamp each frame and send the frame to memory card?
None of the ATmega based boards have the hardware (in terms of acting as a USB host) or power (in terms of handling probably compressed 1080p video).

Potentially one of the ARM boards could do this. The Arduino Due can act as a host, but it sounds fairly experimental, and none of them will have drivers to connect to the camera.

If a shield was developed to do this, it would have so much more power than the Arduino that the Arduino would probably better be viewed as a slave to the other device.