Allinson was the eldest son of a doctor, Thomas Allinson, whose advocacy of vegetarianism and contraception had led to his being struck off, and a German Jewish portrait painter. After leaving Wycliffe College, Allinson began studying medicine, but gave this up and turned instead to art, gaining a scholarship in his second year at the Slade School of Fine Art. Graduating in 1910, he travelled to Europe to study in Paris and in Munich. Following his first exhibition, at the Alpine Club Gallery, in February 1911, he became one of the founding members of the Camden Town Group, and with other members later joined with the Vorticists to form The London Group. 

A pacifist, Allinson associated himself with the Bloomsbury Group during the First World War, producing drawings for the Daily Express newspaper and one of his most important works, a scene inside the Café Royal made in 1915-16. He also designed sets for the Beecham Opera Company. Following the war he again travelled to Europe. 

He became a member of the Royal Society of British Artists in 1933 and of the Royal Institute of Oil Painters in 1936. 

During the 1930s he made a series of posters for London Transport and for the Empire Marketing Board. He was selected as a government war artist by the War Artists Advisory Committee during World War II. After the war, he taught at the Westminster Technical Institute.