William Bramley Jackson obtained the degree Master of Science (MSc) and in July 1904 was appointed assistant inspector of explosives to the government of the Cape Colony. Following the formation of the Union of South Africa in 1910 he was appointed assistant inspector of explosives to the Union government in 1912, and promoted to chief inspector of explosives in 1926. In 1932 he was appointed as a member of a commission of enquiry into a railway accident on 17 July that year, between Leeudoringstad and Wolmaranstad in the western Transvaal. In 1948 he was appointed as a member of the liquor licensing board for the Krugersdorp district.
Jackson was an amateur astronomer. In 1918 he served on the first committee of the Johannesburg Astronomical Association and delivered a lecture to its members that same year. When this association amalgamated with the Cape Astronomical Association in 1922 to form the Astronomical Society of South Africa he was elected as joint vice-president of the latter for 1922/3 and remained a member of its council until 1930. He also served as the first chairman of the society's Johannesburg Section, from 1922 to 1926. The society furthermore formed a Lunar Section under his leadership in 1922, but it failed owing to a lack of observers. He was no longer a member of the society by 1938.
In 1915 Jackson married Olive Maria Palmer, with whom he had a daughter.