Gentlemen - SIDs are designed for the all-engine case. If an aircraft loses an engine it will not necessarily be able to make the alt. constraints in the SID - it depends on the conditions on the day.
Every airline has different procedures for what to do on an engine-failure on departure. In my mob, SOPs state that if the engine fails on the runway or before the initial turn in the SID, we will continue on runway heading. Once we've started a SID-mandated turn we will continue on the SID.
Of course we will inform you nice guys in the Tower ASAP (probably by forgetting all RT procedure and screaming "help"
), but it may well take us a minute or 2 to get around to it as there are other things on our minds at that stage of the game.
Hope this helps.