Nesbitt's inequality was posed as problem 15114 in The Educational Times. According to an online database of the publication maintained by Western Carolina University, it was published in May of 1902. A collection of selected problems and solutions from this publication can be found in Constance I. Marks (ed.), Mathematical Questions and Solutions from "The Educational Times," with many Papers and Solutions in addition to those published in "The Educational Times". New Series. Volume III. London: F. Hodgson 1903. On pages 37 and 38 one finds the problem (proof of the inequality) with three solutions, one of them by Nesbitt himself. On page 3 of the publication he is listed as
Nesbitt, A.M., M.A.: Melbourne
Doing a Google search on this information immediately led me to a website "A biographical register of Australian colonial musical personnel–N" by Dr. Graeme Skinner (University of Sydney) that provides detailed bits of biographical background on Nesbitt (including a photo), confirming information already identified by Dave L Renfro in comments.
Alfred Mortimer Nesbitt was born on December 27, 1854 in England and died July 3, 1926 in East Malvern, Victoria, Australia, which is a suburb of Melbourne. His parents were the physician Pearce R. Nesbitt, M.D. and his wife Arabella. Nesbitt studied mathematics at the University of Oxford, graduating in 1876. In 1877 he was appointed mathematical master of Manchester Grammar School. He married in 1880, as reported in The Medical Times and Gazette, Vol. 2, London: J. & A. Churchill 1880, p. 227:
NESBITT-HILL. — On August 5, at Hampstead, Alfred Mortimer Nesbitt, M.A., of Corpus Christi College, Oxford, youngest son of P. R. Nesbitt, M.D., to Margaret, only daughter of the late John Cartwright Hill, Esq.
In 1882 he was selected as head master of the Grammar School in Toowoomba, Queensland, Australia. The Manchester Grammar School Magazine, Vol. 10, No. 67, October 1882, p. 38:
The cause of athletics in the School has suffered a severe loss by the departures of Mr. H. C. Barnes-Lawrence, M.A., to Giggleswick Grammar School, and of Mr. A. M. Nesbitt, M.A., to the Headmastership of Toowoomba School, Queensland. [...] Mr. Barnes-Lawrence came with our High Master in January, 1877, and Mr. Nesbitt in September of the same year.
In 1888 he became an assistant lecturer of mathematics and natural philosophy at Trinity College in Melbourne. The Melbourne University Calendar 1889-90, University of Melbourne, p. 362 lists him as:
College Teaching Staff.
Mathematics and Natural Philosophy.
(3) A. M. Nesbitt, M.A., late Scholar of Corpus Christi College, Oxford; First-class Honorman in Mathematics at Graduation, 1876.
In addition to being a mathematician, he appears to have been a competent musician and composer. He won first prize in the Jubilee Ode Competition of the Brisbane Musical Union, which presumably was set up for Queen Victoria's Golden Jubilee, which was celebrated in June 1887.