Newcome was born in Newcastle-on-Tyne, England as the eighth of fourteen children. His parents were William Lister Newcombe (1817-1908) and Eliza Jane (Rymer) (1816-1888), both from New York.
He received his MB from the University of Aberdeen in 1873 and his MD in 1878. He married Marian Arnold (1858-1891) in 1879.
In 1884 he established a general practice in Hood River, Oregon. He moved in 1885 with his family to Victoria, British Columbia. His wife Marian died after the birth of their sixth child in 1891, leaving him with two daughters and four sons.
With his eldest three children he returned to England and participated in geological and natural history studies at the British Museum and the University of London. In 1889 he moved back to Victoria and worked at the "Insane Asylum" in New Westminster. He ended to practice medicine after 1894 .
Newcombe began to interest himself in the botany of North America and made many trips to the Queen Charlotte Islands by boat. In the process he became very interested in the Haida Native Americans and started to collect their artifacts to preserve them from the demise of the native culture.
Newcombe and others were driven by the "fear that 'pure' Northwest Coast cultures were disappearing through depopulation and assimilation". In 1897 George Dorsey asked him to collect Haida artifacts for the Museum in Chicago. He also acquired many totem poles for the Royal British Columbia Museum, the Pitt Rivers Museum in Oxford, the British Museum, Kew Gardens in Cambridge, and museums in Liverpool and Sydney. In 1904 he went with six Vancouver Island Native Americans and their magician medicine man to the World's Fair held in St. Louis to show their crafts and culture.
He also conducted biological and geographic research, such as on local (British Columbia) mollusks and paleontology. In 1913 he led a Commission studying the effect of sea lions on the salmon industry. In 1914 he prepared a report on the circumnavigation of Vancouver Island. Much of his work, including collection of plants, mollusks, fossils, aboriginal artifacts and information, was done with the help of his youngest surviving son, William Henry Arnold Newcombe (1884-1960).
He died in 1924 in Victoria, British Columbia after catching a cold on a sailing expedition.