Susskind was born in Prague. His Austrian father was a music critic and his Czech mother was a piano teacher. At the Prague Conservatoire he studied under composer Josef Suk, the son-in-law of Dvořák. He later studied conducting under George Szell. Early in his career, he was often known as H. W. Süsskind (H for Hans or Hanuš).
Susskind fled Prague on 13 March 1939, two days before the German invasion. With the help of a British journalist and consular officials, he arrived in the United Kingdom as a refugee. He formed the Czech Trio, a chamber ensemble in which he was the pianist. Encouraged by Jan Masaryk, the Czech ambassador in London, the trio obtained many engagements.