I understand that all of the members of the British Cabinet are technically appointed by the Queen on the (never ignored, always followed) advice of the Prime Minister. Are they fired the same way, or can Tony himself just personally say, “Hit the road, jack”? Once the Prime Minister decides a Cabinet minister should be booted out of office, what officially has to happen?
A related question: if a Cabinet minister resigns, does her letter to that effect go to the Prime Minister or to the Queen? Do both of them have to do something, or just the Prime Minister?
Politically, the Prime Minister determines who will be Ministers and what portfolios they will have. Each individual is then commissioned in that role by the Crown (the Queen in the UK, the Governor-General here).
If a Minister resigns, stuffs up and decides to “fall on his sword”, or otherwise becomes a sufficiently embarrassing political liability for the Government to want to get rid of him, then again, politically, that’s the decision of the Prime Minister. But the “departing” Minister would submit his resignation to the Crown.
Technically it is the Queen who dismisses ministers. However she always acts upon the advice of the PM. If Blair wanted to suddenly fire a minister he’d have to send a letter to the Queen (or at least phone) “advising” her to dismiss the minister.
Of course, in a coalition government, a PM would have to work it out ahead of time with his partners, unless he wanted the coalition to collapse.