Born in Sheffield, Veitch attended Huntsman's Gardens School before working as an insurance clerk.
[1][2] In 1945, Veitch was elected to Sheffield City Council, serving while continuing to work as a clerk.
[1][2] At the 1951 United Kingdom general election, she stood in Leeds North West, taking second place with 37.5% of the vote.
[2] She resigned from both the council and her work as a clerk in 1956, to take up the full-time post of Yorkshire District Organiser with the National Union of General and Municipal Workers.
[1] Veitch joined the executive of the Confederation of Shipbuilding and Engineering Unions in 1962, the only woman to serve on it in this period.