How did the Reformation affect government in Europe and England?
The Protestant Reformation radically shifted the power relationship between Church and State in the Latin West. The best way to understand this is to see it as part of an historical trajectory.
When the Christian church began, it was a religion of an oppressed minority, outlawed by the Roman Empire. That shifted in 323 when Constantine made Christianity the official religion of the Empire. In the Greek East, the Orthodox Church was an integral part of the Byzantine power structure, in a system sometimes called "Caesaropapism".
As the Latin west fell to the barbarians, civic rule became weaker and more fragmented, with the Roman Catholic Church and its Pope emerging as a uniquely stable power center, often functioning almost as a kingmaker. As the west recovered from the fragmentation of the middle ages and nations became stronger, many rulers began to chafe against the power of the papacy. Protestantism allied with the growing irredentist model of nationalism, in which rather than a loose affiliation of powerful feudal nobles, a nation was considered an integral ethnic, religious, and cultural entity.
The particular issue which most pitted the Roman Church against the powerful monarchs of emerging nation states was that of "church temporalities", the vast wealth and property owned by the Roman church, which was often exempt from taxation. While the French addressed this in a movement known as Gallicanism, which subordinated the Roman Catholic church in France to secular authority (e.g. the French king choosing bishops), for many other European nations, especially England, Protestantism was a way to free the nation from the power of the Papacy (which was actually a temporal power as much as a sacred one) and regain control over the vast wealth and organization of the Church.
An important point to note is that much of what we now consider civil service, such as registering births, marriages, and deaths and administering poor laws, was under the control of the Church, and thus that Protestantism meant, essentially, nativizing one's civil service by wresting control of it from a potentially hostile foreign power (the Papacy was aligned with Spain which was an enemy of England, e.g.)
The Protestant Reformation changed the political landscape of Europe and England by weakening papal authority over secular rulers. For example, the English King Henry VIII declared himself head of the English church in 1534 by passing the Act of Supremacy. While Henry's Act of Supremacy was later repealed, it demonstrated a change in the attitude rulers took toward the relationship between church and state.
The Protestant Reformation also altered government in mainland Europe. The Reformation brought a number of religious wars between Catholics and Protestants. As various princes allied with either the Catholics or Protestants, power became increasingly decentralized.
Aside from the immediate political ramifications of the Reformation, the movement brought another lasting change to government: the principle of questioning authority. The Protestants challenged the authority of the Catholic Church. This principle influenced later Enlightenment thinkers, who began to raise questions about the nature of government and authority.