He was born in Loughborough on 31 October 1832, the son of Reuben Weldon and his wife, Esther Fowke. Weldon was brother to Ernest J. Weldon, founder of Weldon & Wilkinson Ltd. In 1854 he began work as a journalist in London with The Dial, and in 1860 he started a monthly magazine, Weldon's Register of Facts and Occurrences relating to Literature, the Sciences and the Arts, which was later discontinued. In the 1860s he turned to industrial chemistry, described below. However, he is remembered for his pattern work. His publications in the late 1800s were through Weldon & Company, a pattern company who produced hundreds of patterns and projects for numerous types of Victorian needlework. Around 1888, the company began to publish a series of books entitled Weldon's Practical Needlework, each volume consisting of the various newsletters bound together with a cloth cover and costing 2s. 6d. Weldon's Ladies' Journal supplied dressmaking patterns, and was a blueprint for subsequent 'home weeklies'. In 1877 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Alexander Crum Brown, Sir James Dewar, John Hutton Balfour and Sir Andrew Douglas Maclagan. In 1882 he was further elected a Fellow of the Royal Society of London. Weldon was interested in parapsychology, and was a spiritualist and a member of the Society for Psychical Research. He died at Rede Hall near Burstow in Surrey on 5 November 1914.
Weldon was a successful chemist and developed the Weldon process to produce chlorine by boiling hydrochloric acid with manganese dioxide. MnO2 was expensive, and Weldon created a process for its recycling by treating the manganese chloride produced with milk of lime and blowing air through the mixture to form a precipitate known as Weldon mud which was used to generate more chlorine. Manganese dioxide reacts with hydrochloric acid to form chlorine and Manganese chloride: This was put into operation about 1869, and by 1875 it was being used by almost every chlorine manufacturer throughout Europe. He continued to work at the production of chlorine in connection with the processes of creating various sodium salts and became a leading authority on the subject. None of his later proposals met with equal success.