During post-production it is going to be necessary to artificially change the lighting and saturation of some of the shots, both for the sake of making them more interesting, and for continuity (in that lighting is consistent with the shots before and after it).
I had a shot at playing with some of the lighting in iMovie (it was either that or Pinnacle...) to see how easy it's going to be to 'rescue' some of this footage. Results:
(Original shot on the left, edited version on the right)
This shows that it is indeed possible to make a dull grey shot into something which is as colourful as a Gordon Ramsay programme. It is possible to enhance the colours and lighting further (which may be necessary in the second shot from the video) however I have avoided this as it is important to keep a certain degree of realism for the film to work.