Collective cabinet responsibility is the guiding principle of the UK cabinet; it means that if the Govt. comes out with a policy then the whole cabinet must publicly support it, if they agree with it or not. If they choose to speak out about it then they must resign from the cabinet. The main threat to this is the fact that PM's recently have been making decisions with an unelected group of advisors known as the Downing Street Staff.
This decisions could really go against what the actual cabinet feel about the subject, there was a clear example of this recently when Jack Straw had to go on camera and say that it was a good idea to go to war with Iraq, even though according to reports he had spoken to Blair the night before telling him that he thought that we should not go to war. Many feel he only did this to keep his job on the cabinet. Others did not agree and decided to walk out, such as Robin Cook.
Another threat is that ministers may realise that they cannot speak out about the PM and so they 'leak' documents to the press which state that for example a certain minister feels this way on a topic. Leaks were a big problem for John Major during his time as PM due to the fact that he had a hugely divided party which was reflected in his cabinet. He even had to go as far as having a 'back me or sack me' campaign in which he resigned as leader of the Tory party and then again went into the running to see if his party were behind him, due to the amount of leaks in his cabinet, mainly thought to be from Michael Heseltine, among others.
The dominance of a PM is also a major threat to the idea of collective responsibility. This is a PM who would be a force in making basically every decision for the cabinet and give them no part at all in major decisions. A very good example of a dominant PM is Margaret Thatcher who by the end of her time as PM had been so dominant over her cabinet that they decided that they no longer had confidence in her ability to run the party. As the cabinet is representative of the party there was a leadership election which she was well advised not to run in as she would have lost.
This was after the cabinet had been treated as practically a non-entity for many years as Thatcher made them announce her policies and back her publicly. Tony Blair has also been a fairly dominant PM but has recently stated that he is going to work more with his cabinet as he may have seen what lay ahead of him if he did not. Examples of his dominance have been revealed in Robin Cook's memoirs, on topics such as the Dome and the Iraq war. This looks badly on the party as the public feel they as a whole are incompetent.
"Cabinet Govt. has been replaced by Govt. by Core executive". What does this mean? Through the years the UK has been run using the cabinet govt. method in which the cabinet has been involved greatly in making decisions. Recently, excluding the Major years, PM's have used them less and started using other bodies to do more of the jobs that cabinet used to do. This method is known as core executive as the PM is in the centre of them all rather than as in cabinet govt. where there is only one other body. Examples of this are how Tony Blair uses e. g. his Downing Street Staff to make many decisions and would consult them rather than the cabinet. This takes some power away from the cabinet.
As I mentioned earlier Major had to refer back to a cabinet govt. compared to Thatcher's core executive as the party was greatly divided on many issues and so he HAD to give the cabinet a larger share of decision making power so as not to aggravate one side of the party on certain issues, such as Europe. This tied his hands as PM which is partially why his party was seen as weak, as he himself was constantly against half of his party.
In Blair's case he has been able to lead his part due to the fact that they are largely together. When a PM has the ability to lead his party he has the option of switching to the core executive method of decision making. He has surrounded himself with many bodies to broaden his powers. He surrounds himself with such bodies as Think Tanks and the Downing Street Staff who create policies and advise him, rather than the cabinet. Blair also has to listen to such people as Gordon Brown as they have apparently made a deal together as they are two big guns within the party.
So it is true to say that cabinet govt has been replaced by the core executive but this is not new to British politics. The main factor in deciding which method is used is largely down to the PM's personality and circumstances as leader. But on a whole many PM's do use some or all of the bodies involved in the core executive method, but the more used the more obvious it is, I think the more bodies used to make decision by the PM the more power he himself loses as he has less personal control over what is happening within each body.