For a standard approach, the lobby and stair would only need venting. The purpose of the ventilation is to keep the stair clear, not the corridor.
Unfortunately, to be code compliant there would be no need for any make up air in the corridor (unless we go down the road of extended travel distances etc etc), as the stair would provide this to the lobby, being that it is the stair we want to keep clear. Where a standard BRE shaft approach is used, there is never sufficient replacement air for the system to be effective, for any kind of quick clearance. However, it appears to be sufficient for keeping the stair clear according to BRE.
I would question the positioning of the smoke detector. Although I can maybe see some argument to it, the sooner we can get the vents open, the better.