The Inspector is right. When I had this work done ALL doors opening on to the staircase and hallway, including the downstairs doors, had to be replaced.
"The only internal doors opening onto staircases that may have glazing which is not fire resisting and do not need to be fire resisting are doors into bathrooms or toilets."
"The condition of existing doors and door frames must be assessed. If they do not provide at least 20 minutes fire resistance, they must be replaced with doors capable of achieving at least 20 minutes fire resistance."
http://tinyurl.com/ywvcav
As far as I know, existing doors can be 'fireproofed' but it is horrendously expensive and they will look different, so it is quite pointless.