London was born in Bonn in a liberal Jewish-German family. His father, Franz London, was professor of mathematics at the University of Bonn and his mother, Luise Burger, was the daughter of a prosperous textile manufacturer. His father died of heart failure when Heinz was nine years old. The greatest influence on Heinz's childhood was his older brother Fritz. Throughout their lives the two brothers maintained a close relationship.
Heinz followed in his older brother's footsteps, studying physics, but became an experimental physicist instead and obtained his PhD under the famous superconductivity physicist Francis Simon.
This connection also gave Heinz the opportunity to leave Nazi Germany. Frederick Lindemann invited Francis Simon to join the Clarendon Laboratory at the University of Oxford in 1933 supported by money obtained from chemical company ICI. When Francis Simon did he brought Heinz London as his assistant as well as Nicholas Kurti.
While working in Oxford, Heinz shared a rented house with his brother Fritz and sister in-law Edith where together the brothers developed the London equations.
By 1936 the money that had funded the refugee scientists had dried up and Lindemann could not find funds to offer positions to them and many others. Heinz was in a junior position without any expectation of remaining at Oxford, and so took an appointment at the University of Bristol. Fritz held out for a position at Oxford which never came and later accepted an offer by the Henri Poincaré Institute in Paris, in September 1939 moved to the Duke University After the outbreak of World War II.
In 1940 Heinz was declared as "hostile foreigner" and interned for some time on the Isle of Man, but was then released to co-operate with the British nuclear program. In 1942 he obtained British citizenship.