Moor went to Holt High School, Liverpool. His first poem was published in The Listener when he was thirteen. In 1945, when he was 18, he had a long poem on the Liverpool Blitz broadcast. Also in 1945, he went to Cambridge University where he studied English and German. He was awarded the Chancellor's Gold Medal for English verse.

When he was 19 in 1946 he was given the Oscar Blumenthal Prize by Poetry Magazine (Chicago) which meant that he received a small award. After university he was a school teacher in North Wales.

He spent some years abroad teaching English, in particular Japan, Saudi Arabia, and Iran. He used his experiences in Saudi Arabia in The Heat of the Sun, (1976) and Bowl of Roses, (1978). He also used his experiences in Japan in The Pole and the Whistle, (1966), stories with a homosexual theme.