Is it possible to build a 1080p OSD webcam shield?
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.