The Legislative Assembly has published a handy guide to what’s involved in the replacement of a Chief Minister.
What happens when a Chief Minister resigns?
From the Companion to the Standing Orders:
The Chief Minister vacates the office:
— when he or she resigns the office;
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— when the person ceases to be a Member of the Assembly, other than because of a general election—for example, by resignation, disqualification or due to a dissolution of the Assembly; or
— immediately before a Chief Minister is elected after:
— the next general election; or
— the passing of a resolution of no confidence in the Chief Minister.
Although the terms of office of Members cease on the polling day of a general election, the executive continues in office until immediately before the election of a Chief Minister after the declaration of the poll.
The Chief Minister may resign office as Chief Minister by written notice delivered to the Speaker (Neither the Self-Government Act nor the standing orders offer any guidance as to exactly when a resignation takes effect. For practical purposes it is assumed to be from the moment of receipt of the letter of resignation by the Speaker
If a vacancy occurs during a sitting of the Assembly, the election of a new Chief Minister takes place immediately. At any other time, the Speaker must convene a meeting of the Assembly (by notice published in the Territory Gazette) as soon as practicable and, at that meeting, must report the vacancy to the Assembly as soon as possible and the Members present must forthwith proceed to the election of one of their number to be the Chief Minister.
So now we know.