Gwen Burbidge commenced nursing training at Royal Melbourne Hospital, taking first place in eight of the ten requisite examinations in 1929, and receiving the Madge Kelly Memorial Prize for the best nurse in the State. In 1933 she was recruited to the Alfred Hospital as Sister Tutor, where she set up and ran a Preliminary Training School (PTS) for probationer nurses. The school opened on time, though without a single textbook. Miss Burbidge instituted programs of lectures and practical demonstrations, and persuaded the local auxiliary to adopt the School, into which equipment, English texts and a model patient (quickly named Ella Wood) were introduced. Old guard antagonism and resistance mounted as the PTS nurses reached their ranks and old ways and practitioners were pronounced inadequate, but the school won out. Miss Burbidge became Senior Sister Tutor, and in 1934 wrote the first Australian text-book on nursing, Lectures for Nurses, published and distributed Australia wide. It became the core teaching text-book of the day and during World War II it was provided to all medical orderlies in the Australian Armed Forces.

In 1935 Miss Burbidge passed with credit the Victorian Teachers College teaching course and Psychology at the University of Melbourne, and embarked for London for post-graduate study. She was there appointed Assistant Matron at St Thomas's and St James's Hospitals, and Second Assistant Matron at the Manchester Royal Infirmary. She was awarded the First Class Certificate for Sister Tutors, the Sister Tutors' Diploma and the Diploma of Nursing by the University of London. In 1938 she was elected a Fellow and Deputy Chairman of the Board of the Royal Victorian College of Nursing.
Miss Burbidge returned to Melbourne at the end of 1938 to take up the post of Matron of the Queen's Memorial Infectious Diseases Hospital, Fairfield, the first Australian hospital to deal exclusively with infectious disease. Fairfield Hospital at that time was an object of fear, regarded by the public as rather like a leper colony. Patients were delivered to the ambulance from back doors. Families did not visit the patients, whose only contact with people was with nurses swathed in masks and prophylactic uniforms. Loneliness and psychological distress inflicted enormous damage on patients, particularly the young. Nor were staff immune from the effects of the prevailing atmosphere. Matron Burbidge changed the treatment of the patients, the status and condition of the nurses and public attitude to infectious diseases. Twenty years later Sir Herbert Olney, Chairman of Fairfield Hospital Board of Management recorded her establishment of the Nurses Training School, her development of post graduate courses in infectious diseases and her introduction of public health courses. Sir Herbert recorded also Matron's introduction of nurses' graduation ceremonies, training films, occupational therapists, dieticians, specialist theatre units, mobile beds, modern equipment and public education. Uniquely for an infectious diseases hospital, routine visiting hours were established. Matron Burbidge introduced facilities for nurses of a kind later regarded as essential, including drying cupboards, clothes-washing machines, hairdressing facilities, recreational and private meeting areas. Stock control, inventories and hospital supply requirement routines were introduced, public address systems and labour efficient wards created. By 1948 the incidence of infectious disease in Victoria had improved to such an extent that the Act by which the hospital was constituted required amendment to permit vacant wards to be used for the treatment of general patients.

During the war years, Matron Burbidge was elected a member of the Royal College of Nursing of London. In 1942 she was appointed to chair a national conference to report upon the Control of Nursing in War. She was elected a member of the Nurses Registration Board of Victoria in 1944 and in 1946 was appointed by the Australian government to its Committee on Post-war Nursing Reconstruction.

In 1946 she was elected the inaugural President of the National Florence Nightingale Memorial Committee of Australia, serving in that office until 1948. During this time the Committee began to focus on establishing post-basic further educational opportunities for nurses in Australia. The Committee under Miss Burbidge's leadership strongly supported the establishment of a national college of nursing. Miss Burbidge chaired a subcommittee to bring this event about. The goal was achieved in 1949 with the establishment in Melbourne of the College of Nursing, Australia. Miss Burbidge was elected a Fellow of the College and became its first Censor-in-Chief, a position she held until 1960.

Miss Burbidge was Australia's representative at meetings of the Grand Council of the Florence Nightingale International Foundation in 1947, and Australia's representative to the International Congress of Nursing in Washington DC. In 1948 she was appointed to advise the Victorian Government on the formation of the Victorian Hospital Commission.

In 1948 she became the first Australian nurse to receive a Rockefeller Foundation Fellowship, which allowed her to study nursing administration in the United States and Canada. In 1949 she was appointed Australian Representative to the International Council of Nurses and the following year received the Florence Nightingale Award for contribution to the nursing profession. In 1953 she was awarded the Queen's Coronation Medal. In 1954 she was appointed to the Nursing Committee of the National Health and Medical Research Council of Australia and was elected a Councillor of the Royal Victorian College of Nursing. For many years she served as the President of the Airdrie Home for Aged and Incapacitated Nurses. In 1955 Miss Burbidge was awarded the Order of the British Empire for outstanding services to nursing. In 1960 she was elected the President of the College of Nursing, Australia.