Webster graduated from the University of Auckland with a degree in chemistry in 1972, and then completed a PhD in biochemistry, titled Interrelations between purine and pyrimidine metabolism in man, at the Guys Hospital Medical School, King's College London in 1981.

[2] In 1991 Webster became involved in the neonatal screening programme when she was appointed as director of the National Testing Centre at the Auckland District Health Board.

[2] Webster oversaw the policy discussions and other changes required to add severe combined immune deficiency to the screening programme.

Other changes during her leadership included the introduction of a text message alert system for midwives, and improvements to speed and equity of screening.

[3] In the 2020 New Year Honours, Webster was appointed a Companion of the New Zealand Order of Merit for services to health, particularly paediatrics.