An interesting question. If the PM is temporarily incapacitated, is there any way a Minister could be appointed or dismissed?
Technically, these powers are held by the Crown. But, by strong convention, they are exercised only on the PM’s advice. So the obvious answer to the question is “no”.
What, though, if a Secretary of State or Minister dies, resigns, or becomes obviously unable to function, while the PM is incapacitated?
One pragmatic answer is that no one Minister is essential to running any aspect of Govt. Almost all legal powers held by Ministers can be exercised by any of them. So even if a vacancy occurs, there is no immediate need to fill it (eg the Minister of State can step up).
More difficult would be a situation where a Minister clearly had to resign (eg, to take a recent non-Ministerial example, because she decided to drive for the weekend to her holiday home) but refused to do so.
I suppose, ultimately, if the rest of the Cabinet advised HMQ that the Minister had to go, the Queen would be entitled/required to dismiss her. And the threat to involve the Queen would itself operate to put irresistible pressure on the Minister to resign.
But these are uncharted waters, and others may have different views.
