At our airline we have the following general guideline that applies to all aircraft. That's Boeing, Airbus and Embraer.
1. PF - Maintain Aircraft Control
2. Identify the Non-normal
PM Cancel the Warning
3. Accomplish Immediate Action Items
5. Accomplish the Checklist
4. Captain Assigns PF
A flap fault would not likely be an immediate action or memory item. Further explanation of the above policy, outlined in our Flight Operations Manual, would require getting the aircraft to a safe altitude, determine if it is an immediate action item or memory item. Captain assigns the pilot to be flying and then the flying pilot calls "ECAM ACTION". That is the signal to start the ECAM process.
If the ECAM is an immediate or memory item then it is dealt with by "immedate action of the ECAM" or by memory - ie electrical fire. The ECAM for fire can take many different forms and so there is a non ECAM memory list to back up the ECAM system.
Hope this helps.
When in doubt follow your company procedures I am sure they will have good guidance for you.