Do airports have their own incident/accident investigation departments?
More often than not, the accident investigation boards in Europe are a governmental unit independent of NCAA. The Board should be able to criticize and put requirements on all units involved in an accident/incident. This obviously include the CAA. 

Departements and organizations are free to do an accident investigation. Availability to an accident site may be limited, though.
E.g. IFALPA have a structure of their own Accredited Accident Investigator, available for any relevant investigations.