Cosslett was born in Cirencester in June 1908, the son of Edgar William Cosslett and Anne Cosslett (née Williams). He was awarded a PhD at Bristol University in 1932, and was then appointed a Research Fellow at Bristol, 1932-5; Lecturer in Science, Faraday House, London, 1935-9; research (part-time) at Birkbeck College, London, 1936-9; Keddey-Fletcher-Warr Research Fellow of London University (at Oxford), 1939-41; and Lecturer in Physics, Electrical Laboratory, Oxford University, 1941-6.

The majority of his career, however was spent at the Cavendish Laboratory in Cambridge and he was appointed successively: ICI Fellow, Cavendish Laboratory, 1946-49; Lecturer in Physics, University of Cambridge, 1949-65, Reader in Electron Physics, University of Cambridge, 1965-75, and then Emeritus. He was a Fellow of Corpus Christi College, Cambridge, from 1963.

He was a founder member of the International Federation of Electron Microscope Societies, and its President from 1972-5. He was also President of the Royal Microscopical Society and a Vice-President of the Institute of Physics. He was elected a Fellow of the Royal Society in 1972 and received its Royal Medal in 1979.

He published "Introduction to Electron Optics", 1946 (1951); "Practical Electron Microscopy", 1951; "X-ray Microscopy" (with W. C. Nixon), 1960; and "Modern Microscopy", 1966.

He died in November 1990.