To my knowledge, except on TERPS, where on normal departure you have 0.8% margin on the said departure ( besides ATC constraints or specific airports location); since PAN-OPS 2 and above there is no provisions for OEI departure. SID are just airspace 3D tracks with a minimum performance designed in them.
If an airplane loses an engine, then, the operator has to provide an acceptable alternative option, and this for every take off, every weight, every intersection of every airport intended to be used. This track can well be the opposite of the normal departure ( i.e. GVA Left turn in OEI instead of Right Turn for SID).
As far as eu-ops is concerned, no turn are allowed below 50ft, between 50 ft and 200ft max bank to 15° up to 400ft max bank 20° up to 1500ft max 30° If banking more than 15° obstacle clearance goes to 50 ft vertically. Again some airport designs ( AJA/GRZ/...) might require something different, but it is definitely not something that the PIC choose at the last minute.
In FAA land, the distance increases with distance 48ft/Nm....0,8%....
This is from memory... so..