The son of a nurseryman, Randall was born at Newton-le-Willows, then in Lancashire, and educated at the University of Manchester. He began work with General Electric at their Wembley laboratory, originally seeking to develop luminescent powders. He was subsequently appointed in 1937 to a Royal Society Fellowship at Birmingham University to conduct research into luminescence.

But the coming of World War II stopped all such nonmilitary work and Randall, along with his Birmingham colleagues, began work on improving current radar techniques. By 1943, in collaboration with his colleague Henry Boot (1917–1983), he had developed the cavity magnetron, one of the most vital inventions of the war.

Radar operates by reflecting radio waves off objects. As only a tiny part of the transmitted radiation is reflected back, and as only a fraction of this could be picked up by the receiving antena, considerable power had to be transmitted. Moreover the precision with which a target's position could be located was a function of the narrowness of the transmitted beam. Early transmitters had operated with wavelengths as long as 15 meters spreading the signal over a 100° sector. The demand was for ‘centimetric radar’; i.e., radar using wavelengths of a centimeter or two.

The answer came with Boot and Randall's cavity magnetron, first revealed in November 1939. The device consists of a block of copper with accurately machined cavities through which circulating electrons subject to a magnetic field would generate intense electromagnetic radiation. When tested the magnetron delivered 400 watts on a 9-cm wavelength. The magnetron was passed to the research department of General Electric who increased its power to 10 kW. The new magnetron radar was first used by Bomber Command but soon became available to fighter aircraft and convoy escorts.

With the end of the war in sight Randall was appointed professor of natural philosophy in 1944 at St. Andrews in Scotland. In 1946 he accepted the post of head of a new department of biophysics set up by the Medical Research Council at King's College, London. Under him were Maurice Wilkins and Rosalind Franklin, two scientists intimately connected with the elucidation in 1953 of the structure of DNA.

Randall himself worked on the structure of collagen, an important protein giving the skin its elasticity; it was found to consist of three helices coiled into a superhelix. On his retirement to Edinburgh in 1970 Randall continued to work with an informal research group on problems in biophysics.