One of the stranger consequences of Queen Elizabeth II’s death is that a good chunk of the British government now exists in a sort of constitutional limbo. Although she approved numerous ministerial appointments right before her death, she was unable to preside over a Privy Council meeting to swear in the new Cabinet.
I’ve written about the byzantine process of making ministerial appointments elsewhere, but to make a long story short, the Promissory Oaths Order 1939 requires most Cabinet ministers to take the requisite oaths before the Sovereign in Council. There can be other formalities as well. For example, certain ministers receive seals of office from the Monarch, while the President of the Board of Trade is appointed by Order in Council. But, strictly speaking, senior ministers do not formally assume office until these steps are completed.
In practice, however, there seems to be some wiggle room. When David Cameron reshuffled the Cabinet on September 4, 2012, some of the new ministers weren’t sworn in until October 17. Even Prime Ministers have sometimes had to wait: in July 2016, Theresa May didn’t take the oath as First Lord of the Treasury until six days after her appointment. Despite these delays, it appears that everyone was allowed to start work ahead of their formal swearing-in.
Given the many demands on the King’s time, it may be a while before the i’s can be dotted and the t’s crossed. Luckily, while the British have a reputation for being sticklers for protocol, they also have a sense of pragmatism that is helpful in times like these.
 All ministers must take the official oath, and if they haven’t already taken the oath of allegiance, they must take that, too.
 This is a sinecure that’s lately been held by the Secretary of State for International Trade.
 Most junior ministers (i.e., those outside the Cabinet) take office from the moment the Sovereign signs the Prime Minister’s submission recommending their appointment. Even if the late Queen was unable to do that before her death, the King could easily finish the process without much fuss.