This depends on what the processor does. Most consumer level cameras will lose 20%-25% in resolution. That's the reason for the 4K needing the 4.6 sensors in BMD and Canon models. Higher end cameras may double sample (remember the old CD players advertising 2X and 4X? ... this is similar, albeit without the noise shifting) in a different sequence and, some cameras (Alexa XLT and its sisters) will even claim the in-camera upscaling that delivers resolution greater than its stated number of pixels.But surely no 1080 footage from a single chip camera will ever be 1080 - after Bayer you lose approx. 20-30% as far as I have always understood it.
Steve
PS. This is done on consumer cameras too, where the Olympus OMD can triple the stills resolution with the pixel shifting technology. In a $2,000 EM1 MKII model, the pixel shifting works well on the interiors and landscape photography but it is too slow for portraiture. On a $82,000 Alexa SXT, however, the in-body upscaling seems to work well on anything.