Cromwell and his supporters are often described as republicans, and while that makes sense during the civil wars, I want to know if they still thought of themselves that way afterwards. When the Commonwealth became a hereditary dictatorship, did the Roundheads try to draw any distinction between how that was any different from a monarchy?
It's important to note that the parliamentarian alliance that had defeated King Charles I became increasingly fractured as the Civil War came to a close. Many of the groups which had been a part of it were excluded from the new government as events such as Pride's Purge and the dissolution of the Rump and Barebone's parliaments showed. There was really no consensus on what the final political status of England should be, which was one of the major reasons the Commonwealth eventually collapsed. Cromwell himself was fairly committed for both religious and political reasons to at the very least not having a crowned head of state in England.
Yet even his supporters were divided in their religious and political ideals. Eventually Cromwell was offered kingship as part of the "Humble Petition and Advice" in 1657, but he refused. So while Cromwell was personally republican there was really no unified "roundhead" idea of what England should be. There were definitely strong republican sentiments, but others who didn't have a particular attachment to it one way or the other. Generally though any outright monarchist feeling was quite low among those who had actual influence in the Commonwealth's institutions.