Drumm was born in Newry, County Down, to a staunchly Irish republican McAteer family, where she became the eldest of four siblings. Drumm's mother, Margaret McAteer (née Brown), had been active in the War of Independence and the Civil War. Drumm grew up in the village of Killeen, County Armagh, right on the border with County Louth. She played camogie for Killeen. The family moved to Dublin in 1940 and it was not long afterwards that Drumm had joined Sinn Féin. The family moved again to Liverpool and it was there Drumm joined the Gaelic League. The family returned to Northern Ireland in 1943 and Drumm took up work as a grocer's assistant in Belfast. It was at this point Drumm became an active partipant in the Republican movement, taking a particular interest in the welfare of Republican prisoners. It was through this interest she met Jimmy Drumm, a brewery worker interned in a Belfast prison, whom she married upon his release in 1947. During the 1950s and 1960s Jimmy Drumm spent 13 years in prison for Republican activity, leaving Máire to do most of the raising of their four children. Despite her family circumstances, Drumm remained an active social figure. She became involved in the National Graves Association, an organisation dedicated to the maintenance of the gravesites and shrines to deceased Republicans. She was also involved in the Gaelic Athletic Association, working to promote the sport of Camogie.