Born at Horton in the Staffordshire moorlands into a Wesleyan family and brought up in Hanley, he came under the influence of an aunt, Elizabeth Dakin, a Primitive Methodist preacher in the Cloud Primitive Methodist chapel at Congleton. He was influenced by Bethesda Methodist New Connexion Chapel, where he and his first wife were married in 1856 and he became a trustee in 1862. He and his second wife were later leading MNC members at Mount Tabor,Fenton and Ebenezer, Newcastle under Lyme, at a time when the Hanley and Longton Circuits were strongholds of the denomination.
Harvey followed his father James into the Pottery trade, joining the MNC Shelley family business in 1861/62. When J.B. Shelley was declared bankrupt in 1862 the saved him from the prospect of imprisonment by paying his outstanding debts. From 1869 to 1885 Harvey Adams and Co. were know for their high quality products, many of them for the export market, including the invention and design of the moustache cup, acknowledged as his invention. The family firm was also associated with other MNC pottery firms, notably that of Titus Hammersley.
Harvey became widely known for his refusal to pay the School Rate introduced in the1870 Education Act, on the grounds that it unfairly favoured the Church of England. He was possibly the first to do so, and was described in the press as the school rate martyr of the Potteries. He sold the family business in 1885 to his partner George Hammersley, but continued to travel in retirement for other potters, the Forester Phoenix works and the Methodists Mayer and Sherratt. As a successful businessman, energetic local politician and loyal to the New Connexion, he reflects much that was best in Victorian Methodism and nonconformity. He died in 1916, surviving his eldest son who had been Medical Officer of Health for Warrington.