Henry Dyer was born on 16 August 1848, in the village of Muirmadkin (now absorbed in to the town of Bellshill) in the Parish of Bothwell in what is now known as North Lanarkshire.
Around 1865, the Dyer family moved to Glasgow where Henry was employed at James Aitken and Company's foundry in Cranstonhill. There he served his apprenticeship as a student engineer under Thomas Kennedy and A C Kirk. At the same time, he attended classes at Anderson's College (later to become the University of Strathclyde) together with Yamao Yōzō.
Dyer studied at Glasgow University from 1868 He was the first Scot to win the Whitworth scholarship awarded in 1868, which was for the further instruction of young men gifted in the practice and theory of mechanics. Henry Dyer graduated from Glasgow University in 1873 with a "certificate in proficiency in engineering", the forerunner of the BSc in Engineering, from the Engineering department.