That is essentially what happened with Nixon, though under different circumstances. Members of his party, Republicans, went to him and told him he would lose an impeachment and conviction vote. " These articles were reported to the House of Representatives for final action, with 7 of the committee’s 17 Republicans joining all 21 of its Democrats in voting in favor of one or more of the articles. Two other articles were debated in committee but were rejected. Based on the strength of the evidence presented and the bipartisan support for the articles in committee, House leaders of both political parties concluded that Nixon’s impeachment by the full House was a certainty if it reached the House floor for a final vote, and that his conviction in a Senate trial was a distinct possibility.
On August 5, 1974, Nixon released a transcript of one of the additional conversations to the public, known as the “smoking gun” tape, which made clear his complicity in the Watergate cover-up. This disclosure destroyed Nixon politically. His most loyal defenders in Congress announced they would vote to impeach and convict Nixon for obstructing justice. Republican congressional leaders met with Nixon and told him that his impeachment and removal were all but certain." It was a different time then, when politicians valued country over party.
A Chinese tanker has sailed past the American blockade in the straits after China issued Trump a warning not to interfere in their affairs. Meanwhile a US warship is sailing the long way around Africa to avoid coming into contact with the Houthis so afraid in coming into contact is going to defend who? If they are scared of the Houthis.
In the event that the Prime Minister had clearly gone crazy, then the King, advised and supported by the Privy Council, would probably quietly advise the PM to resign on grounds of ill health. If that didn’t work, then His Majesty would have two options;
Prorogue Parliament and cause a General Election to take place where the rogue PM would no doubt lose his seat in the Commons even if he was still supported by his Constituency Party, which would be unlikely.
Dismiss the PM from his post and ask someone else to form a Government in the King’s name.
Although it is traditional for the Monarch to appoint the leader of the party with the largest presence in the Commons as PM, there is no actual requirement for that to be the case. The Monarch can ask anyone they choose to form a government. One must remember that at the end of the day the British Government is not the winning party’s government, nor even Parliament’s government, but is in fact The Monarch’s Government, appointed by The Monarch, to govern in his / her name.
So a truly crazy PM could fairly easily and quickly be dismissed from their post by the person that appointed them in the first place, namely The Monarch.
It’s a very different, and much safer system, than having a directly elected President.
I hope this goes at least some way to answering your question.
The safeguard there is that the PM is not the Commander-in-Chief of His Majesty’s armed forces. Technically the PM holds the position of First Lord of the Treasury, which is what is on the brass plate on the door of 10 Downing Street.
An MP can be deselected by their local party/branch/constituency, and if that MP is PM then… It’s a route. In Labour’s case, the NEC can pull the plug too. The Tories, the 1922 committee (?)
Returning to Trump bashing he’s now slagging off the UK because “we weren’t there” when he needed help with his illegal war and is threatening to rescind the UK-US trade deal. He really is a piece of sh*t.
Not so - the most recent one was Sir Alec Douglas-Home who became PM while still technically a member of the House of Lords (as Lord Home), but he resigned his peerage immediately on becoming PM. He then had to stand for election to the House of Commons but that happened pretty quickly I think.
The Marquess of Salisbury who was Prime Minister on and off between 1895 and 1902 was the last PM before that to hold the office while not a member of the House of Commons, though he was an MP before his father’s death brought him his hereditary peerage.
The last PM who had never been an MP and only sat in the Lords was Lord Rosebery (1894-95).
In theory the PM can be anybody appointed by the King, but Google says it’s never happened that they have not been a member of either the Commons or the Lords.
Then the growing voices of keep Plantir out of the UK health service has a perfect route to not go there either. Screw America down, let the other countries unite so we can get rid of the terrible regime some Americans have to live under.
I think that at some point, some countries are going to start thinking about whether trying to work with America is more painful than simply turning their backs on them and forging new alliances. If that sort of thinking catches on then America and the rest of the world could be in for a very painful time. Yes, it would be a no win situation either way, but it may be a case of least worst.
Not saying this would happen, but trying to work with an increasingly unhinged President and an increasingly dysfunctional, unpredictable and irrational U.S. government could create a turning point in the relationship between the U.S. and a big chunk of the rest of the world.
As a fellow pedant, I was about to post that.
. Good job there are some here who are glad to help out 
