Why did the United Kingdom scrap Conscription in 1960?.
When all the Other European countries did not scrap Conscription until the 1990s, the end of the Cold War?.
The short answer is the 1957 Defence Review, called by others on this site "The Sandystorm."
The longer answer is that the British armed forces relied on voluntary enlistment until 1916. It was abolished at the end of World War One and re-introduced in the spring of 1939 to provide the personnel needed for the doubled Territorial Army field force and the massive expansion of Anti-Aircraft Command. Conscription for all the armed forces was introduced when World War II broke out and this time was continued into peace. Initially the conscripts had to serve 2 years, then as part of the austerity cuts down to 18 months and when the Korean War broke out it was put back up to 2 years.
By the late 1950s the cost of maintaining HM Forces was absorbing (IIRC) 10% of the UK's GNP. This was not bankrupting the country but it was a severe financial strain. In an era of full employment the large standing armed forces (and the expanded arms industry) were depriving the civilian economy of badly needed workers. However, the final straws were the advent of the hydrogen bomb and the ICBM which to the British Government made the maintenance of large standing armed forces and equally large reserves to fight a conventional war in Europe pointless.
Therefore Sandys invented what became known as the "East of Suez" strategy. The forces being maintained to fight World War III in Europe would be gutted (see the severe cuts to BAOR, the Territorial Army, RAF Germany, Fighter Command and Coastal Command) leaving only Bomber Command relatively untouched because it was the nuclear deterrent force. The rump of the Army, RAF and Royal Navy would be restructured to fight counter insurgency wars in the Third World before they escalated into bigger wars.
The smaller armed forces required less manpower, which could be recruited by voluntary enlistment, so no need for conscription.