For "last possible minute" substitute "at the appropriate place in the single-engine approach checklist". Put simply, not until the aircraft is established on its normal descending glidepath. This is because you dont want the remaining good engine to be spooled-up to the extent necessary to overcome the drag of the undercarriage, flaps, etc in level flight before final descent is commenced.
It's the "nurture carefully what you have got left" philosophy which every self-respecting professional pilot will subscribe to. True, the remaining engine can produce enough power to effect a go-around by itself, but why "cock the trigger" unnecessarily?