Gascoigne's parents met and married in Levin, New Zealand, just before the First World War. They soon moved to Napier, where Gascoigne was born in 1915. He attended Auckland Grammar School, and won a scholarship to Auckland University College (now the University of Auckland) a year before he was due to finish high school. Faced with a choice between studying history or the sciences, he chose the latter because he had a severe stammer and thought that it would be less of an impediment. He completed both a Bachelor's degree and a Master's qualification in science, securing Honours in both mathematics and physics, finishing his studies in 1937. Despite these achievements, he did not consider himself to be practically trained, saying: "I was still very much a theorist, with no practical physics at all. The professor in Auckland used to wince when I walked past the cupboard in which the good instruments were kept!"
