Basically, as mentioned by Kurnal, the premises has to be suitable for a stay put policy. In general this would entail 60 mins compartmentation (If there is a floor above 5m) between each occupiers dwelling, 30 minute fire doors to corridors, possibly ventilation to stairs/corridors or lobbies and a suitable alarm system. If this is in place then a stay-put is probably valid, and generally safer than potentially evacuating people through smoke. Any evacuation done by the FRS would be at their discretion, not down to any specific policy.
Look at it from this point of view: If the compartmentation etc is in place then people are quite safe in their flats, if the necessary measures for stay put are not in place, then by delaying the evacuation you are putting people at more risk.
I have seen it too many times where a RP is aware that it is difficult to evacuate all the people that should be evacuated and they see 'stay put' as an easy way out. A policy does not protect people in their flats, it is really compartmentation/fire resistance that allows them to stay there.
With regards the stairs question, it is unlikely that a stair in a residential building will not meet the flow requirements, can you give any more detail on numbers and staircase widths?