Allinson was born in London, his mother was a portrait painter and his father a doctor. He originally studied medicine before studying at the Slade School of Fine Art; fellow students included Nevinson, Stanley Spencer and Edward Wadsworth. Allinson exhibited with the London Group, the New English Art Group and at the Royal Academy. In addition to painting, he became chief scenic designer for the Beecham Opera Company, designed posters for the railways, the Empire Marketing Board and taught at the City of Westminster College. During the 2nd World War he was appointed an official war artist, contributing work for the Dig for Victory campaign. A retrospective exhibition was held at the Fine Art Society, London in 1984.