The British draft was ended as a result of the 1957 defence white paper which itslef was a result of the loss in the Suez war. It was decided that Britain was no longer an extra European power and therefore would scrap its big army and air force and defend itself with nuclear missiles.
So your PoDs are victory at Suez and/or some brains in the 1957 white paper. I still think Britain would ditch the draft, but perhaps a decade later on.
Actually no the were three simple reasons why National Service went when it did.
1. Too many people. The whole point of National Service was that it was universal, everyone did it unlike the draft. However if you look at demographics by 1964 you are beginning to get the massive baby boomer generation. The Armed forces didn't want all these people, they couldn't afford to be expanding the army and didn't need to.
2. Sufficient volunteers. Apart from a few backbenchers no one supported National Service out of principle, it was a solution to a problem, in 1946 Britain had a needed a lot of soldiers to do a lot of jobs and could not afford to hire them on the job market. By the 1960's you could, the Army could meet its manpower needs purely through volunteers.
3. Technical Sophistication. In the early years of National Service there were NS pilots, which was possible when it took 6 months to learn to fly a Typhoon. By the '60's it took two years and half to train a Lightning pilot the new aircraft were so complicated. While the Army didn't have the same degree of problem as the RAF or RN it had similar issues.
National Service was very much a short term solution to a short term problem and if any government had been elected on a platform of bringing it back they would have faced the stringent opposition of the entire Military hierarchy who didn't want it back.