The republic was officially known as the Commonwealth of England, Scotland and Ireland. The period from 1653 to 1659, when the Cromwells were (successively) afforded the title of Lord Protector, is now better known as the Protectorate.
The Commonwealth period is often referred to as the Interregnum. For obvious reasons, Wikipedia presents this as the Monarchist viewpoint. Elsewhere, Wikipedia describes 'Interregnum' as a generic term for "a period of discontinuity or 'gap' in a government, organization, or social order. Archetypally, it was the period of time between the reign of one monarch and the next" – of which the Commonwealth of England (1649–1660) is just one example.
© Haydn Thompson 2017–21