At least 8 days before your intended arrival date. Miss that and a £50 late fee applies, and applications submitted 3 days or fewer before arrival will not be approved at all. Because Antarctic itineraries move with weather and ice, coordinate the timing with your cruise operator rather than working it out alone.
If you've missed the deadline, contact your operator or agent immediately. Don't assume you can board and sort it out after sailing, because a missing permit affects whether you are legally authorised to land.
The permit runs 40 days from its commencement date, and the government advises choosing dates that leave room for the itinerary to shift. That flexibility is there for a reason on this route.