John, or "Jack", Heslop-Harrison was born in Middlesbrough to John William Heslop-Harrison, a botanist, and Christina Henderson. He attended the Elizabethville Elementary School and Chester-le-Street Secondary School, where he was awarded a scholarship to attend King's College, Newcastle in 1938 to study chemistry, zoology and botany. He graduated with first-class honours in 1941- as did his future collaborator and wife, Yolande Massey.

He saw service during the war; initially given a place on a radio operator course, as a result he spent some of his remaining time at the university doing a course at the physics department on electronic wave theory He was trained to operate radio equipment in relation to radar and geolocation, and towards the end of the course also got to handle the then-new cavity magnetron. He graduated first in his course and chose to be posted to Orkney. As part of his technical work he was a frequent visitor to the Royal Electrical and Mechanical Engineers (REME) base on the Orkney mainland; was appointed operator of the complex and promoted to captain. His war service was uneventful, though late in the war after the beginning of the V2 attacks he took worked on developing radar capable of tracking their trajectories.

In March 1945 he was posted to Brussels, with the mission of retrieving a sample of the fungus Eremothecium ashbyi from the Dutch National Mycological Collection at Baarn; it had proved useful in synthesising vitamin B, something in demand in post-war Europe. After the war he was assigned to teams tasked to retrieve technological data from German research facilities. At Pelzerhaken, near Denmark, where many of the scientists uprooted by allied bombing had been based, Heslop-Harrison's team found research into infra-red detection, radar systems and U-boat signature masking.

After his army service he worked briefly in industry, on penicillin production, but soon left to become a junior lecturer at King's College (then Armstrong College) at the University of Newcastle (then part of the University of Durham). A year later (1946) he moved to Queen's University, Belfast as a lecturer in botany. His principal interests were in the physiology of flower development - most especially orchids - and exogenous controls of sex expression in plants; he also completed a PhD in just two years, entitled "Studies on flowering and the phenomenon of sex intergradations and reversal in some angiosperm families". He also lectured and took tutorials. Outside Queen's, he was energetically involved with the Belfast Natural History and Field Club, was editor of the Irish Naturalists' Journal, and was invited to write a comprehensive synopsis of the dactylorchids of the British Isles, at the request of the International Phytogeographic Excursion organisers, later published.

Heslop-Harrison later singled out as one of his most seminal experiences at Queen's the visit to Ireland of the first post-war International Phytogeographic Excursion in 1949, when he not only came into contact with distinguished specialists in plant geography - in taxonomy, phytogeography, ecology, genecology and palynology. - but especially W. H. Pearsall FRS, who immediately offered him a lectureship in University College London to commence in 1950, the post being specifically designated for taxonomy. He was promoted Reader in 1953, in which year he published his groundbreaking and internationally-successful book, New Concepts in Flowering Plant Taxonomy, a work which represented a marked departure from traditional approaches, dealing with micro-evolution, speciation, phenotypic plasticity, plant geography and reproductive isolation, the use of cytological methods and the taxonomic and evolutionary significance of cytological observations.

Heslop-Harrison (he would begin to use this hyphenated form to avoid confusion with another Professor Harrison at Queen's, with whom he was often confused by the internal mailing system) returned to Queen's in 1954 as Professor of Botany to take up the challenge of modernising a small and not well-funded department. The Vice-Chancellor by this time was another eminent botanist, Eric Ashby (later Baron Ashby FRS). Heslop-Harrison had overlapped with him for one term before leaving for London in 1950; he and Ashby were both interested in how the onset of flowering is regulated. Heslop-Harrison wrote him a detailed letter setting out a hypothesis on photoperiodic control of sex expression in plants to which Ashby responded by providing space in a greenhouse at the Vice-Chancellor's Lodge near the university, there being no adequate on-campus departmental facilities at that time. He also was keen to be able to observe Heslop-Harrison's work at close quarters.

In his second period at Queen's he designed a new Botany Department in a new science building; he built up the departmental staff by appointing first-rate lecturers in physiology and biochemistry, ecology and plant geography, and he established a Quaternary Research Unit which developed a flourishing programme on Irish vegetation history through carbon dating and palynology. He published 12 experimental papers (more than half co-authored by his wife Yolande) and 25 other articles, adding physiology to his already wide range of expertise which included experimental taxonomy, biometrics and phytogeography.

He left Queen's again in 1960 to the University of Birmingham, which was in the process of unifying its various biology departments in one School of Biological Sciences, something he oversaw, becoming Mason Professor of Botany in 1963. In 1967 he was awarded the Trail-Crisp Medal by the Linnean Society of London, and the same year became the first Chairman of the Institute of Plant Development at the University of Wisconsin-Madison, having already spent a sabbatical nine months there in 1965. In 1970 he was formally offered the position of Director of the Royal Botanic Gardens, Kew. He spent a year as "Director designate" without official duties or pay, spending much of this time researching for his position, and writing an acclaimed book-length account of sexual reproduction in flowering plants as well as a landmark paper with botanist Bruce Knox tracing the fate of protein diffusing from pollen grains on to stigmatic papillae, notable for their introduction of the term "recognition substances". He also wrote a forward-looking article entitled "The Prospect Ahead" in the Journal of the Kew Guild, which featured his list of the functions of the Royal Botanic Gardens: public amenity and education; curation of the largest existing collection of living plant species, exceeding 20,000 taxa, and the compilation of floras and monographs; operation of a seed distribution system and a seed bank; horticultural training through a diploma course; identification, advisory, quarantine and forensic services for various clients world-wide; research in plant taxonomy, geography, anatomy, physiology, cytology, biochemistry, ecology, genetics; plant introduction and many aspects of horticulture and associated technology. Government often added the duty of entertaining visiting dignitaries. The Royal Botanic Gardens also had its own detachment of police.

However, although he made wide-ranging and widely-applauded changes to the way the institute worked he nevertheless had differences with the government, who funded the institute, and eventually resigned in 1976, the first Director to do so since the position was created in 1822. After leaving Kew he was accepted a position as Royal Society Research Professor at the Welsh Plant Breeding Station (WPBS, currently the Institute of Grassland and Environmental Research), Aberystwyth University, becoming thoroughly engrossed in research; from his departure from Kew onwards he published 106 papers. In 1985 he retired as Research Professor due to the age requirement, but both he and Yolande were made Honorary Visiting Workers.

Heslop-Harrison was awarded numerous medals, awards, degrees both honorary and other, and distinctions, served on many boards of scientific societies, and gave many named lectures throughout the world. In 1953 he was elected a Fellow of the Royal Society of Edinburgh, the first of many national academies to give him such recognition; in March 1956 he was appointed a Member of the Royal Irish Academy, with the citation "Recognised authority on certain aspects of the flora of Britain and Ireland, especially the orchid flora". In 1970 there followed election as a Fellow of the Royal Society; other such distinctions included the Koninklijke Nederlandse Botanische Vereniging in Holland, the Indian National Science Academy, the American Botanical Society, the Deutsche Akademie der Naturforscher Leopoldina, Germany, the American Academy of Arts and Sciences, and the Académie Royale de Belgique, Belgium.

In 1970 he received an honorary DSc from Queen's University, Belfast, something that he later said he especially prized due to his involvement there. However, he would, along with some other honorary graduates, relinquish it in 1995 in protest at a certain policy of the University which he regarded as objectionable and of which he was extremely critical. Newcastle University awarded him a DSc for his published work from 1951-1959. In 1982 he was awarded the Darwin Medal of the Royal Society jointly with his wife, Yolande.