How did England's politics affect America's in the 1400s-1600s?

The phrasing of the question--how did England's politics affect America's in the 1400s to 1600s--is a bit tricky, insofar as the first English settlement in North America was established at Jamestown, Virginia, in 1607. To discuss the affects of England's politics on America prior to that period, therefore, is a bit of a misnomer. Putting aside the rather inconsequential nature of Sir Walter Raleigh's adventures in North America, the English political imprint on the future United States of America resided primarily in England's export to "the New World" of its Protestantism, its martial tradition, and the political philosophies of Thomas Hobbes and, more importantly, John Locke. Hobbes and Locke both proved influential in the intellectual development of Britain's North American colonies. While the Reformation encouraged rejection of the autocratic dictates associated with Roman Catholicism, Locke's philosophical writings provided the intellectual underpinnings for the founding documents of the United States, mainly, the Declaration of Independence and the Constitution. In his Second Treatise of Government, for example, Locke wrote:

" . . .the consent of the people, which is the only lawful basis for government" (Preface to the Two Treatises)

“Being all equal and independent, no one ought to harm another in his life, health, liberty, or possessions.” (Chapter 2: The State of Nature)

" . . .whenever the legislature breaks this fundamental rule of society and—whether through ambition, fear, folly or corruption—try to grasp for themselves or for anyone else an absolute power over the lives, liberties, and estates of the people, by this breach of trust they forfeit the power the people had put into their hands for quite different purposes." (Chapter 19: Dissolution of Government)

Such sentiments, it is clear, were well-represented in the constitutional provisions intended to limit the powers of government and to protect fundamental liberties. Indeed, they form the basis for the democratic system of governance Americans enjoy today. In this respect, English political theory, as well as the obvious experience of living under the yoke of the English monarchy, greatly impacted the emerging nation of the United States. It would be an exaggeration, however, to credit entirely the writings of John Locke, as French political theorists, including Rousseau and Voltaire, were equally influential.

The chronology of American history is such that the time period specified in the question can only lead to confusion. Prior to the settlement of Jamestown, as well as the Massachusetts settlements of the Puritans, there is little evidence of English influence in the Americas. Until those settlements, North America was more of a Spanish and, later, French playground than an English one.