Unless the revolution somehow takes place before 1792, the term "commune" signifying a revolutionary city government has already been done in Paris (if it's after 1870, it's been done twice). Still, more boring terms like "council", "committee", "assembly", etc. probably prevail.
The new British authorities probably want to do anything possible to separate themselves from the legacy of Cromwell, who was and is viewed as a mass murderer of Irish people (fair) who wanted to be a dictator (not really). If they're drawing at all from the English Civil Wars, it's probably from the more radical Levellers. They probably call themselves a Republic rather than a Commonwealth.
The new British authorities probably want to do anything possible to separate themselves from the legacy of Cromwell, who was and is viewed as a mass murderer of Irish people (fair) who wanted to be a dictator (not really). If they're drawing at all from the English Civil Wars, it's probably from the more radical Levellers. They probably call themselves a Republic rather than a Commonwealth.