I understand where you are coming from, but this problem never existed for Cromwell. The House of Commons passed an ordinance in October 1646 abolishing the offices of bishops and archbishops (Kenyon, Stuart Constitution, p. 257; Journals of the House of Commons, 4:688). The Deans and Chapters of churches would be abolished later in 1649. The Episcopacy was the most distinct feature of the Church of England in the years leading up to the Civil Wars, and their abolition meant near total destruction of the English Church (as established by law). That does not mean that suddenly there was no church in England. It just meant that the structure ranging from King to the Archbishop, to the bishops, to the village priests had its head chopped off. What exactly remained is a matter for debate, as men like Cromwell tried to promote a style of Independent churches, where the congregation was the source of power. The Scots had promoted their idea of a Presbyterian system, where each church would have a council of elders who were in charge of the local affairs, but there remained a national church that could decide doctrine (this is an over simplification). Thus, when Cromwell was in power, he was able to fight for religious toleration for various Independent congregations, but before Cromwell had this power, nearly four years had passed since the death of Charles.

Initially, following the death of Charles and the abolition of the Monarchy, England was ruled by the House of Commons, with a Council of State exercising the majority of executive functions. This was the period of the true English Commonwealth, where, until the spring of 1653, the Commons, the Council of State, and the Army held nearly all political power. That is not to say Cromwell was silent or not without authority during this period; he was undoubtedly one of the most powerful men in England. But, he had not been offered the crown, he was not Protector, and much of his time was spent fighting rather than ruling. This was the period when Cromwell invaded Ireland and won his infamous victories at Wexford and Drogheda. It was only in April of 1653 after multiple delays by the Commons to establish a more representative Commons, or at least a fair method of creating such a body, that Cromwell dissolved the sitting parliament at musket-point (Kishlansky, Monarchy Transformed, p. 187-189). Within a year Cromwell had assumed the title of Lord Protector, and would call his first parliament in September of 1654. It was not until after Cromwell's death that the episcopacy would be restored.

Besides the episcopate, another key element of the Church of England was that it was intended to be all inclusive - to be a subject to the King of England meant you were also expected to be a member of the Church of England. Those who refused to attend annual communion were known as Recusants, which typically meant Catholics before 1642. With the breakdown of church hierarchy during the 1640s, it also became harder to compel people to attend the religious services of the "Church of England". Groups of Independents, Presbyterians, and eventually Baptists, Quakers, and Anti-Trinitarian/Socinian groups began to meet. The 1653 Instrument of Government, which established Cromwell as Lord Protector specifically states "that such as profess faith in God by Jesus Christ (though differing in judgement from the doctrine, worship or discipline publicly held forth) shall not be restrained from, but shall be protected in, the profession of the Faith, and exercise of their religion..." This is an intentionally broad definition to include moderate episcopalians, Presbyterians, and Independents (Hughes, "The Cromwellian Church" in Oxford History of Anglicanism, Volume 1). Catholics were specifically excluded from these protections (as well as "prelates" meaning bishops and those they ordained, but in practice this was not enforced).

Besides a wider toleration of what was considered acceptable Christian doctrine, Cromwell created two groups intended to enforce Christina orthodoxy (or something akin to it), and helped maintain public funding for ministers. The two groups intended to enforce some level of orthodoxy were known as the Triers and Ejectors. The Triers was a committee of 38 men who were to approve all ministers appointed in the counties that would receive public money, and was mostly composed of Presbyterians and Independents. The Ejectors were locality based committees intended to remove "scandalous" or poor quality ministers. The public funding for ministers came in largely in the form of the hated tithes, and this provided funding to about 80% of ministers in England. So while Cromwell may not have been the head of the church in England, his powers combined with compliant parliaments allowed him to establish a form of religious toleration, funding for ministers, and a means for punishing and ejecting problematic ministers. Cromwell also adopted, for the most part, the ancient parish divisions that had been around for centuries. He and his council tried in some areas to redistrict the lines of worship so as to provide better funding and administration of the Christian Religion, but much of these developments were little out of infancy at Cromwell's death in 1658.

One last element worth considering is that in Scotland, for those that belonged to the national church, the head of said church was Charles II. There is a lot to say on the status of the national Scottish church(s), but I'll be brief. The Scottish church had prevented most all attempts made by Charles I to instill English doctrine and law, and instead operated as a national Presbyterian church. In order to gain their support for his bid to the throne, Charles II was held hostage aboard a ship and forced to sign the "Solemn League and Covenant" - a statement that bound him to support the Kirk and Presbyterian form of church government, at the expense and eradication of the episcopacy (again, an over simplification) (Hutton, Charles II). Cromwell's conquest of Scotland in 1650-1651 broke much of the political resistance to his later Protectorate, and due to Cromwell's sympathies for Presbyterianism (though he was not personally a Presbyterian) made the national Scottish church less of a problem than the remnants of the English church.