I'm aware Cromwell and other generals had strong personal beliefs, but was that more important to them than secular beliefs like liberty and parliamentary sovereignty? Were some beliefs (Arminian/Lutheran/Episcopalian etc) more associated with certain groups, like the Cavaliers, the Levellers, the Diggers etc) and did this inform the philosophy of their movements more than temporal affairs? Did these beliefs go all the way down from Cromwell to the common soldiery?
The short answer is hugely important and religious beliefs (or just as often, alleged religious beliefs) were often used to ostracise opponents. Religious conflict and the threat of infiltration by Catholics was used to justify extreme measures in parliament, including the arrest and execution of many of Charles I’s advisers by parliament. Both sides would accuse the other of being ‘papists’, ie supporters of the Roman Catholic Church. Pamphlets would be used to deride and ridicule enemies, as well as stirring up fear and making accusations.
In particular, anti-Royalists accused members of Charles I’s court of papist plotting in order to remove key advisors from power. It was a believable story, as Charles I had a Catholic wife, Henrietta Maria, who kept a small Catholic court including a priest. The idea that the royal household had been infiltrated by Catholicism was a very damaging one for Charles. An example of a high profile figure who fell victim to such accusations of popery was Archbishop Laud. Though clearly not a Catholic, Laud was accused of being one due to his support of maintaining certain Catholic rituals in church services (his beliefs were broadly Arminian).
The accusation of being a Catholic was often fatal. Laud was beheaded for treason in 1645. The Duke of Buckingham, one of Charles’s closest advisors and friends, was assassinated in 1628, following two attempted impeachments by parliament for treason (charges included that he was negotiating with French Catholics about the overthrow of parliament). When parliament were attempting to obtain the execution of another key adviser of Charles’s, Thomas Stafford Duke of Wentworth, parliament hauled a 73 year old Catholic priest out of a prison cell to give evidence in parliament as to the current and immediate threat of popish plots - setting the trial of Stafford within a frenzy of anti Catholic fear, used to justify his execution in 1645.
I studied this era and wrote a dissertation on the religious aspects of the English Civil War. I focused on reading contemporary propaganda and pamphlets and it was pretty clear that religion was used to dehumanise an enemy group.
Sources- Tessa Watt, Cheap Print and Popular Piety (Cambridge University Press, 1991) Caroline Hibbard, Charles I and the Popish Plot (University of North Carolina Press, 1983) The Thomason Collection, an archive of early modern pamphlets and popular literature.