Maxwell Bury was born on 28 July 1825, at East Retford in Nottinghamshire, England, the son of William Bury, an Anglican clergyman, and his first wife, Harriet Fowler. He spent part of his childhood in Cambridgeshire, where his mother died when he was about two years old. His father subsequently remarried, and the family returned to Nottinghamshire in 1833.
Maxwell trained in engineering at an ironworks near Derby; there are no details of his architectural training. He probably visited Birmingham, as the great Jacobean house Aston Hall, near Birmingham, is thought to have been the inspiration for the Nelson Provincial Government Buildings, one of his first major works in New Zealand. He took up marine engineering, and went to sea as an engineer officer in the merchant service.
On 11 August 1853, at Ellesmere, Shropshire, Maxwell Bury married Eleanor Sarah Deighton (known as Ellen). The following year they sailed for Australia on the Zingari, a small steam-assisted sailing ship which Maxwell Bury had built in partnership with some friends. After several frustrating months in Melbourne, where the economy was in depression, they continued on to New Zealand, arriving at Nelson on 12 December 1854.
Bury contracted with the provincial governments of Nelson and Wellington to run a coastal service with the Zingari. Plans to charter a larger steamer for a connecting Tasman service were not realised, but the Zingari continued the coastal run until 1858. Ellen Bury ran a small school in Nelson, and Maxwell set up as an engineer and land agent, as well as becoming involved in mining and transport ventures. In 1857 he was elected chairman of the first Nelson Board of Works, and in 1861 was appointed sheriff of Nelson.
Maxwell Bury's first architectural commission was probably the extensions to Christ Church (the cathedral). He designed the Masonic Hall, built in 1858, and in August 1859 the foundation stones of the Nelson Provincial Government Buildings and the Nelson Institute were laid. He also designed the Māori chapel at Wakapuaka in 1862 (later relocated, and finally destroyed by fire), and may have been the architect of the early part of Bishopdale, the house begun by Bishop Edmund Hobhouse.
Bury was deeply involved in Anglican church affairs in Nelson. He was a churchwarden and later treasurer of the diocesan synod, the Nelson lay representative at the General Synod, and diocesan surveyor. When he moved with his family to Christchurch in early 1863 he remained active in the church. His architectural work in Christchurch included the Torlesse building in Cathedral Square, an orphanage at Addington, several private houses, and the stone, octagonal-towered Church of St John the Baptist in Latimer Square, designed in early 1864. He served on a committee for workers' housing, and judged the competition for the Christchurch Supreme Court Building with Edward Dobson.
In mid 1864 Bury went into partnership with B. W. Mountfort. Together Mountfort and Bury designed several churches, including St Mark's at Ōpawa, St James' at Cust and St Joseph's at Lyttelton. However, the partnership did not last; in March 1866 Maxwell and Ellen Bury and their two small daughters left for England, possibly at Ellen's insistence, as she believed her mother was dying.
By 1870 Maxwell Bury was back in Nelson, apparently alone, where he resumed his architectural career. In 1877 he designed the Chapel of the Holy Evangelists on the hilltop at Bishopdale. Soon after, he moved to Dunedin after winning the competition for the design of the University of Otago. His design was reportedly classical, but seems to have been changed to Gothic at the university's request. He also designed the four professorial houses in a simplified red-brick Gothic style.
Bury continued his work with the church in Dunedin, but architectural jobs were scarce. With some local doctors he tried unsuccessfully to float a company to develop a spa at Rotorua. He briefly returned to Christchurch where Mountfort was designing Canterbury College. Then in 1883 the University of Otago commissioned the second stage of the science building. Until mid 1885 Bury worked on designs for a university hall, but the scheme was abandoned as too costly and he returned to Nelson.
Sometime after 1890 Bury moved to Sydney, and in 1908 returned to England. He died at Ledbury, Herefordshire, at the home of his elder daughter, on 9 September 1912. His fine watercolour architectural drawings remain in the Canterbury Museum, the Hocken Library and the archives of the Nelson diocese.