Engine failure at take off, you control the yaw with the rudder to maintain the runway centre line and then lock off your leg. Once airborne keep the wings level and make sure you are flying the correct heading.
Any other engine failure scenario, you initially control the roll using aileron, and then apply rudder to bring the control column back to neutral. The end result should be wings level, on a constant heading with the control column neutral. We identify which rudder to press by pushing whichever side of the column is low. Ie, if you need to turn the control column to the right in order to keep wings level, then you push the right rudder until the control column comes back to neutral, you then trim. We are no taught to use the slip indicator on the PFD.
How quickly? Quick enough to keep the runway centre line. Identifying the engine you can use the dead leg, dead engine theory.
SW