Can I use Project Atlas just before the Runner accesses cards from HQ?
After reading the FAQ (here), specifically the section on the timing of a run, I believe I can now answer my own question. The relevant steps read:


  4 The Runner APPROACHES the attacked server.
  
  
    4.1 [Paid abilities]
    
    4.2 The Runner decides whether to continue the run.
        ...Either the Runner JACKS OUT: [end the run unsuccessfully]
        ...Or the Runner continues the run: go to 4.3.
    
    4.3 [Paid abilities and non-ice can be rezzed]
    
    4.4 The run is considered to be successful. (‘When successful’ conditionals meet their trigger conditions)
    
    4.5 Access cards, then [end the run].
  


Based on these clarifications, you may use the ability before or after the Runner decides if they'd like to access.