Second, you need to set the resolution for the movie (Shooting menu, Movie settings, Frame size/frame rate) to '1920x1080' and either '30fps', '25fps' or '24fps'.
Third, you need to set the HDMI output resolution (wrench, hdmi output resolution). This setting goes up to only '1080i' and not '1080p', so set it to 'Auto' to enable the 1080p capability.
Fourth, you need to remove the SD and CF memory cards from the camera. This may sound strange, but when a memory card is present the Nikon D800/D800E will try to record the movie in 1080p to the memory card and the hdmi output will switch back to 720p. Apparently the processor in the Nikon D800/D800E is too slow to handle both recording and outputting 1080p at the same time. This is likely a hardware limitation so don't keep your hopes up for a firmware update.
Plug in your external hdmi video recorder (such as the portable Atomos Ninja or AJA KiPro mini) and have fun. (Note: the Atomos Ninja needs firmware 2.2 or later to accept 1080p from the D800). You can just record the live view, no need to press any other button on the camera. Just use the record and stop buttons on the recorder.
Technical note: the output appears to be 8 bit 4-2-2 color sampling video on the best possible quality setting but these numbers are not mentioned in the official Nikon specs.