Allom attended Wellington College, Berkshire, before going up to Cambridge University. He played for Cambridge University from 1926 to 1928 and for Surrey from 1927 to 1937. He toured with the English Test team to New Zealand in 1929-30, where he played all four Tests, and to South Africa in 1930-31, where he played one Test. He and his former Cambridge team-mate Maurice Turnbull wrote a book about each tour: The Book of the Two Maurices: Being some account of the tour of an M.C.C. team through Australia and New Zealand in the closing months of 1929 and the beginning of 1930 (1930) and The Two Maurices Again: Being some account of the tour of the M.C.C. team through South Africa in the closing months of 1930 and the beginning of 1931 (1931).
Almost 6 feet 6 inches tall, Allom was able to get the ball to rise sharply off the pitch. Along with Peter Petherick and Damien Fleming, Allom is one of only three players to have taken a hat-trick on Test debut. In the same Test against New Zealand, he also became the first player to take four wickets in five balls, a feat later matched by Chris Old and Wasim Akram. He finished with first innings figures of 5/38.
He served as Surrey's President from 1970 to 1977, and as President of MCC in 1969-70. He was also a skilful saxophonist, who played in Fred Elizalde's band in the 1920s.
His son Anthony played first-class cricket for Surrey and was one of tallest people to have played the game, standing between 6 ft 9 ins and 6 ft 10 ins (around 2.07 m).