As a matter of fact Cromwell reported directly to the head of the Church of England.
Cromwell was no Royalist, he was at loggerheads with the monarchy the entire time. He was a staunch Parliamentarian, known more popularly and derisively as a Roundhead. He was a Republican, not a monarchist.
Beginning about 1640, tensions rose between King Charles I and Parliament. Parliament had a problem with Charles raising taxes without their approval. Members of Parliament were troubled to the point of outrage by Charles's association with Catholicism, particularly his queen, Henrietta Maria, who was herself Catholic. Relations between King and Parliament were severely strained, leading to the outbreak of war, the English Civil War, in August of 1642. The Parliamentarian Cromwell raised an army in his own home county, and after a string of victories over the King's armies, was eventually elevated to the rank of Lieutenant General.
Parliament's army eventually defeated the King and his supporters, the Royalists, and Cromwell emerged from the Civil War as an heroic figure in England. By the late 1640s, Cromwell became the de facto leader of the Parliamentarian army, and he consolidated this into one miliary force known as the New Model Army. After Charles I was executed in January 1649, Cromwell's influence and leadership of the military meant he was a very powerful political force, and so he assumed the prestigious new position of first chairman of the new Council of State.
To imply that Cromwell was acting at the behest of the Church of England is ludicrous as a result. He no more acted at church behest than did Winston Churchill.
Compare this to armies raised by the Vatican itself, and sent against the supposed heretic Pietists, Moravians, Albigensians/Cathars and Waldensians ... there is no equivalence whatsoever. That is the distinction with a difference to which I referred.