Baxter Greig was born in was born in Camberwell, London, England on 29 April 1878. After working in the office of R. F Beaumont for six months in 1894, he was articled to Richard John Lovell (1858?-1931) in London. He also attended the Architectural Association Schools and the Royal Academy Schools in London. He qualified as an architect in 1902 and subsequently commenced practice as an architect in London.
By 1919 Greig had been appointed District Surveyor for Deptford. He was later District Surveyor for St. Marylebone East and Interim District Surveyor for St. Marylebone West.
He was elected an Associate of the Royal Institute of British Architects (ARIBA) in 1902 and Fellow of the Royal Institute of British Architects (FRIBA) in 1935.
His address was given as Fifeshire, East Dulwich Grove, Dulwich, London in 1902; 183 East Dulwich Grove, Dulwich, London in 1911; Bank Chambers, 329 New Cross Road, London in 1914 and 1919; 82 Crescent Lane, London in 1926; 1 Newman Street, London in 1939; and 5 Alleyn Park, West Dulwich, London in 1950. He died in Camberwell, London on 7 July 1950
Directory of British Architects 1834-1914. Compiled by Antonia Brodie, et al. Volume 1: A-K. London; New York: British Architectural Library, Royal Institute of British Architects/Continuum, 2001
'Obituary'. The Builder vol. 179, 4 July 1950 pp. 72, 167
'Obituary'. RIBA Journal vol. 57, August 1950 pp. 406-407