Neilson was born in Doune, Scotland on 28 March 1869.[2] He emigrated to the United States in 1895, being naturalised 3 August 1905. He taught at Bryn Mawr College from 1898 to 1900, Harvard from 1900 to 1904, Columbia from 1904 to 1906, and Harvard again from 1906 to 1917. He served as President of Smith College from 1917 to 1939.
Neilson was author of a number of critical works on William Shakespeare, Robert Burns and the Elizabethan theatre, editor of the Cambridge and Tudor editions of Shakespeare (1906, 1911) and editor of Webster's New International Dictionary, Second Edition (1934).[3] Less known is his translation of the famous late 14th-century Middle English alliterative chivalric romance Sir Gawain and the Green Knight.