To use a certain procedure you have to meet the required climb gradients of the procedures with all engines operating.
Actually procedure designers simply design to a gradient (3.3% for a SID and 2.5% for a missed approach, unless otherwise specified) - they couldn't care less how many engines you have operating.
One of the constraints for the missed approach, however, is that it begins at the Missed Approach point & MDA/DA. A previous company I worked for specified that, should you go-around below the minima for the procedure, you were then to follow the engine out procedure, not the missed approach (as you no longer met the design criteria).