Bhatnagar was born in the Bhera, Punjab region of British India, in a Hindu kayastha family. His Father, Parmeshwari Sahai Bhatnagar, died when he was eight months old, and he spent his childhood in the house of his maternal grandfather, an Engineer, from whom he developed a liking for science and engineering. He enjoyed building mechanical toys, electronic batteries, string telephones. From his maternal family he also inherited a gift of poetry. He studied elementary education at the Dayanand Anglo-Vedic High School, Sikandrabad (Bulandshahr). In 1911 he joined the newly established Dayal Singh College, Lahore, where he became an active member of the Saraswati Stage Society. Bhatnagar earned a good reputation as an actor. He wrote an Urdu one-act play called Karamati (Wonder worker), the English translation of which earned him the Saraswati Stage Society prize and medal for the best play of the year 1912. Bhatnagar passed the Intermediate Examination of the Punjab University in 1913 in first class and joined the Forman Christian College,where he obtained a BSc in physics in 1916, and a MSc in chemistry in 1919.
Bhatnagar was awarded a scholarship by the Dyal Singh College Trust to study abroad, and he left for America via England. However, he could not find open berths on English ships, as they were all reserved for American troops, who were then being demobilised due to the First World War. The Trustee permitted him to join the University College London under chemistry professor Frederick G. Donnan. He earned his DSc in 1921. While in London, he was additionally supported by the British Department of Scientific and Industrial Research with fellowship of £250 a year. In August 1921, he returned to India and immediately joined the newly established Banaras Hindu University (BHU) as a professor of chemistry, where he remained for three years. He wrote the ‘Kulgeet’ (University song). Justice N.H. Bhagwati, Vice-Chancellor of BHU said: "Many of you perhaps do not know that besides being an eminent scientist, Professor Bhatnagar was a Hindi poet of repute and that during his stay in Banaras, he composed the ‘Kulgeet’ of the University...Prof. Bhatnagar is remembered with reverence in this University and will continue to be so remembered till this University exists." He moved to Lahore as Professor of Physical Chemistry and Director of University Chemical Laboratories of the University of the Punjab. This career was the most active period of his life in original scientific work. His research interests included emulsions, colloids, and industrial chemistry, but his fundamental contributions were in the field of magneto-chemistry, the use of magnetism for the study of chemical reactions. In 1928 he and K.N. Mathur jointly invented what they called the Bhatnagar-Mathur Magnetic Interference Balance, which was one of the most sensitive instruments for measuring magnetic properties. It was exhibited at the Royal Society Soiree in 1931 and it was marketed by Messers Adam Hilger and Co, London.
Bhatnagar persistently refused personal monetary benefit from his research fundings, and instead advocated for strengthening research facilities at the university. His sacrifices drew wide attention. Meghnad Saha wrote to Bhatnagar in 1934 saying, ‘You have hereby raised the status of the university teachers in the estimation of public, not to speak of the benefit conferred on your Alma Mater’.
For his excellent contributions to pure and applied chemistry, Bhatnagar was appointed an Officer of the Order of the British Empire (OBE) in the 1936 New Year Honours List. He was knighted in the 1941 New Year Honours List for his contributions to the advancement of science. In 1943 the Society of Chemical Industry, London, elected him as Honorary Member and later as Vice President.Bhatnagar was elected a Fellow of the Royal Society (FRS) in 1943.
The first industrial research organisation in India was created as an Industrial Intelligence and Research Bureau, which came into operation in April 1935 and under the Indian Stores Department. With its limited budget of Rs. 1.2 lakh per annum, the bureau was virtually inactive. In 1939 there was a strong movement to abolish the bureau, and to replace it with a Board of Scientific and Industrial Research, similar to the British BSIR. Under the persuasive pressure of Arcot Ramaswamy Mudaliar, the Board of Scientific and Industrial Research (BSIR) was formed on 1 April 1940 for a period of two years. Bhatnagar, as a leading scientist of the time, was appointed as the Director, and Mudaliar became the Chairman. The BSIR had an annual budget of Rs. 5 lakhs, which was placed under the Department of Commerce. Bhatnagar persuaded the government to set up an Industrial Research Utilisation Committee (IRUC) in early 1941 for further investment into industrial research. Mudaliar also won the demand for an establishment of Industrial Research Fund, and that it should have an annual grant of Rs 1 million for a period of five years, at the Central Assembly in Delhi at its session on 14 November 1941. These finally led to the constitution of the Council of Scientific and Industrial Research (CSIR) as an autonomous body, which came into operation on 28 September 1942. The BSIR and IRUC became the advisory bodies to the governing body of the CSIR. In 1943 the governing body approved the proposal mooted by Bhatnagar to establish five national laboratories — the National Chemical Laboratory, the National Physical Laboratory, the Fuel Research Station, and the Glass and Ceramics Research Institute. This was the beginning of scientific laboratories in India.
Bhatnagar played a significant part along with Homi Jehangir Bhabha, Prasanta Chandra Mahalanobis, Vikram Sarabhai and others in building of India's post-independence science and Technology infrastructure and policies. Prime Minister Nehru was a proponent of scientific development, and after India's independence in 1947, the Council of Scientific and Industrial Research (CSIR) was set up under the chairmanship of Dr. Bhatnagar. He became its first Director-General, and by his works he is largely remembered for having established various chemical laboratories in India. He established a total twelve national laboratories such as Central Food Processing Technological Institute, Mysore, National Chemical Laboratory, Pune, the National Physical Laboratory, New Delhi, the National Metallurgical Laboratory, Jamshedpur, the Central Fuel Institute, Dhanbad, just to name a few. While at CSIR, he mentored a number of promising young Scientists of the time who were working in the very productive climate in those days at the Indian Association for the Cultivation of Science (IACS) in Kolkata, including Syamadas Chatterjee, Santilal Banerjee (MSc Gold Medalist- Dacca University and a DSc from the US) who later moved to the National Physical Laboratory in Delhi at Bhatnagar's urging), Asutosh Mookherjee etc. Bhatnagar also closely followed the work of C. V. Raman and Kariamanickam Srinivasa Krishnan at IACS on the Raman Effect and in particular the work Krishnan and his partner, Santilal Banerjee, were doing on the magnetism of small crystals.
He also served as Secretary of the Ministry of Education and Educational Adviser to government. He played an important role both in the constitution and deliberations of the Scientific Manpower Committee Report of 1948. ‘It may be pointed out that this was the first-ever systematic assessment of the scientific manpower needs of the country in all aspects which served as an important policy document for the government to plan the post-independent S&T infrastructure.’ Bhatnagar was a university professor for 19 years (1921–40) first at the Banaras Hindu University and then at the Punjab University and he had a reputation as a very inspiring Teacher and it was as a Teacher that he himself was most happy. His research contribution in the areas of magnetochemistry and physical chemistry of emulsion were widely recognised. He also did considerable work in applied chemistry. He played an instrumental role in the establishment of the National Research Development Corporation (NRDC) of India, which bridges the gap between research and development. Bhatnagar was responsible for the initiation of the Industrial Research Association movement in the country. He constituted the one-man Commission in 1951 to negotiate with oil companies for starting refineries and this ultimately led to the establishment of many oil refineries in different parts of the country. He induced many individuals and organisations to donate liberally for the cause of science and education.
He died of a heart attack on 1 January 1955, aged only 60.