When Gwynn was ten years old, the family moved to Llangefni, because one of her father's siblings had found employment as Cardiff's Regional Dental Officer for Wales. She was taught at Pwllheli County School, and was attracted to chemistry, mathematics and physics due to influential teachers and her father. Gwynn defied the school's predominant English-language education, and assaulted an English mistress, who called the girls in the class "Welsh scum". In 1930, she joined Plaid Cymru, five years after the party was founded. Gwynn earned a scholarship to read physics at the University College of North Wales in 1934, after declining to attend the University of Cambridge. She was the only woman in her class to study physics at the college, and received abuse from the external examiner because she was a woman. Gwynn graduated three years later with a higher second class degree.