Abel was born in London, the son of a well-known musician and the grandson of a court painter to the grand duke of Mecklenberg-Schwerin. Despite this artistic background, Abel developed an early interest in science after visiting his uncle A. J. Abel, a mineralogist and pupil of Berzelius. In 1845 he was one of the first of the pupils to study at the Royal College of Chemistry under August von Hofmann, remaining there until 1851. After a brief appointment as a chemical demonstrator at St. Bartholomew's Hospital, London, he succeeded Michael Faraday in 1852 as a lecturer in chemistry at the Royal Military Academy at Woolwich. In 1854 he became ordnance chemist and chemist to the war department.

Abel's career was thus devoted exclusively to the chemistry of explosives. New and powerful explosives, including guncotton and nitroglycerin, had recently been invented but were unsafe to use. Abel's first achievement was to show how guncotton could be rendered stable and safe. His method was to remove all traces of the sulfuric and nitric acids used in its manufacture by mincing, washing in soda until all the acid had been removed, and drying. In 1888 he was appointed president of a government committee to find new high explosives. The two existing propellants, Poudre B and ballistite, had various defects, most important of which was a tendency to deteriorate during storage. Together with Sir James Dewar, Abel introduced the new explosive, cordite, in 1889. This was a mixture of guncotton and nitroglycerin with camphor and petroleum added as stabilizers and preservatives.

Abel was honoured for his services by being made a knight in 1891 and a baronet in 1893.