The 1968 Plot
In his 1976 memoir Walking On The Water, Hugh Cudlipp recounts a meeting he arranged at the request of Cecil King, the head of the International Publishing Corporation, between King and Lord Mountbatten. The meeting took place on May 8 1968. Attending were Mountbatten, King, Cudlipp, and Sir Solly Zuckerman, the Chief Scientific Adviser to the British government.
According to Cudlipp:
"[Cecil] awaited the arrival of Sir Solly and then at once expounded his views on the gravity of the national situation, the urgency for action, and then embarked upon a shopping list of the Prime Minister's shortcomings...He explained that in the crisis he forsaw as being just around the corner, the Government would disintegrate, there would be bloodshed in the streets and the armed forces would be involved. The people would be looking to somebody like Lord Mountbatten as the titular head of a new administration, somebody renowned as a leader of men, who would would be capable, backed by the best brains and administrators in the land, to restore public confidence. He ended with a question to Mountbatten- would he agree to be the titular head of a new administration in such circumstances?" 
Mountbatten asked for the opinion of Zuckerman, who stated that the plan amounted to treason and left the room. Mountbatten expressed the same opinion, and King and Cudlipp left. On 30 May 1968 King was dismissed as the head of the International Publishing Corporation.
It should be noted that in addition to Mountbatten's refusal to participate in King's mooted plot, there is no evidence of any other conspirators. Cudlipp himself appears to see the meeting as an example of extreme egotism on King's part.
 A Military Coup In 1974?
On the BBC television programme The Plot Against Harold Wilson, broadcast on March 16, 2006 on BBC2, it was claimed there were threats of a coup d'état. Wilson told two BBC journalists, Roger Courtiour and Barrie Penrose, that he feared he was being undermined by the MI5. The first time was in the late 1960s after the Wilson Government devalued the pound sterling but the threat faded after Conservative leader Edward Heath won the election of 1970. However after a coal miners strike Heath decided to hold an election to renew his mandate to govern in February 1974 but lost narrowly to Wilson. There was again talk of a military coup, with rumours of Lord Mountbatten as head of an interregnal administration after Wilson had been deposed. In 1974 the Army occupied Heathrow Airport on the grounds of training for possible IRA terrorist action there, however Baroness Falkender (an intimate friend of Wilson) claimed that it was ordered as a practice-run for a military takeover.