Twomey, first elected to the Newfoundland assembly in 1985, served stints as Minister of Health from 1984 to 1988, and as Minister of Public Works and Services from 1988 to 1989.
He represented the electoral district of Exploits in the Newfoundland and Labrador House of Assembly from 1976 to 1989 as a member of the Progressive Conservative Party of Newfoundland and Labrador. 
Twomey was first elected to the Newfoundland assembly in 1975. 
