The standard reading of Henry VIII and the New World is pretty unexciting. Henry VIII was certainly aware of the discovery of the new continents at the start of his reign, and he may have known the extent of wealth pouring into Spain from the discovery. However, Henry did not have much interest in the New World, and seems to have devoted little thought and fewer resources to pursuing projects in the Americas.

Instead, Henry focused most of his time, and nearly all of his wealth, to issues in France and the continent. Henry, like many English Kings, styled himself the King of France by birth right, and he sought to regain his lost inheritance by force. Almost all of his military exploits to regain some title to French land failed miserably, and Henry went through much of his newly acquired monastic wealth fighting wars in France. Henry sought the glory of victory on the battlefield. He studied Henry V's battles in France, and often compared himself to one of his namesakes. Henry even focused more on fighting France than concerning himself with the often hostile kingdom at his rear - Scotland. Though Scarisbrick in his biography of Henry VIII claims that Henry had ideas of a united Great Britain, later works have shown that Scotland was an afterthought for Henry, and more a kingdom to be placated than a place of conquest (Doran, England and Europe).

Henry's father Henry VII did have an interest in exploring the new world. The most famous example was the Cabot voyages of 1497. John Cabot was born in Italy, but moved to Bristol sometime c. 1495. Henry VII presented Cabot with a patent to "to sail freely in search of new lands to the east, west, and north (excluding Spanish claims to the south)" (ODNB Online, "John Cabot" by David Quinn). Evan Jones has also published recently that there were further expeditions in the late fifteenth century out of Bristol intended to explore new lands. Cabot likely encountered Newfoundland off the coast of Canada, and there was a persistent presence of English fishermen in the North Atlantic for the next few centuries. The best work on this is probably still Quinn, England and the Discovery of America, 1481-1620.

In reality, it does make sense that Henry VIII was more focused at home than abroad. Aside from his disputed claim to the French throne, Henry was orchestrating massive religious, political and institutional reform in his little archipelago. Henry instituted the parish system in Wales in 1541, and ensured that the "King's writs could run" everywhere in his dominion. Further, he proclaimed himself Lord of Ireland, and expended no small amount of effort trying to bring that island under his control. Henry also, with the assistance and direction of Wolsey, Cromwell, and More (at their respective times), dissolved the monasteries, made himself Supreme Head of the Church on earth, and attempted to reform the church's theology and practice on the ground. Whether one considers Henry's actions positive or negative, there is no doubt that England under his reign experienced massive changes in the nature of authority, belief, and finance.

When you say "the other tudors", I assume you mean "apart from Elizabeth"? Because if not, then yes, Elizabeth did have explorers such as Drake and Raleigh make expeditions to the new world.