Hi Longo,I think OzExpat and others have coverered what you asked.
In both OEI as in AEO,you may follow the SID,but terrain clearance has not been assessed for climb gradients of less than 3.3%.The min required net gradients of climb for 2,3 and 4 engine A/C with OEI are less than this,so you would need to assess the surrounding terrain during your briefing.You can then decide on your best option .There may be no terrain problems and you may choose as PIC to follow the SID following an engine failure.
Major airlines will cover all situations , but some of the smaller or less reputable outfits will leave everything up to the PIC.At the end of the day,you are responsible for the safety of the aircraft and passengers.
The SID departure is based on an average 15 degrees of bank and a maximum speed of 290 Kts,so the OEI speed of roughly 120-160 Kts up to 290 Kts,gives a fairly wide speed band.
Anyway,you have made me start to think about again,so good luck in the future.