We have a list of things that are mandatory to report including airport, pilots deviating from a clearance, icing conditions, heavy turbulence, anything where a Pan or Mayday is declared.
It's unlikely the actual controller will get involved in speaking to the aircrew as all our reports get passed to the Investigations department and they take it from there. We can get feedback on the results when they get them.
As for timescales, it depends entirely on the incident and what the Investigations lot need to find out or who they need to contact. UK airlines are pretty good although trying to track down of the bizjet pilots can be more difficult so any reporting will take longer. Obviously it's all got to be run past the CAA at some point
This is from a UK perspective and in particular NATS en-route