Richardson, the son of a farmer, was born in Newcastle-upon-Tyne and educated at Durham College and Cambridge University, graduating in 1903. In 1913 he became superintendent of the meteorological observatory at Eskdalemuir, Scotland. His work was interrupted by World War I and he resigned from the Meteorological Office in 1920. He was head of the physics department at Westminster Training College, London, until in 1929 he became principal of Paisley Technical College, Scotland, where he remained until his retirement in 1940.

Richardson was the first to try to apply mathematical techniques to weather prediction, publishing his ideas in Weather Prediction by Numerical Process (1922). In this he argued that the state of the atmosphere is defined by its temperature, pressure, and velocity. Once these were known, he believed that equations could be used to predict future weather conditions. The main problem with implementing his program was the time taken for computation, and it also suffered from a shortage of information. This was partially resolved with the advent of electronic computers following World War II.

The Richardson number, a value involving the gradients of temperature and wind velocity, is named for him.

Richardson also attempted to apply a mathematical framework to the study of the causes of war, publishing his work in Generalized Foreign Politics (1939), Arms and Insecurity (1949), and Statistics of Deadly Quarrels (1950).