Similar questions have been posted before, but England Discovering the New World is still a very fascinating idea. First of all, OTL Columbus sent his brother Bartholomew to England 1489 to search for support. Meanwhile, the same year Columbus began receiving an allowance from Spain in return for Columbus not offering his ideas elsewhere. Therefore, the POD would be England accepting Bartholomew Columbus' offer in 1489, which place Columbus' voyage in 1490 or 1491. As for Vespucci, it seems like he went where the action, and money, were, so if Columbus discovers America for England, Vespucci is probably close behind.
As for what route Columbus would take, it is hard to tell. On the one hand, he understood the Atlantic Equatorial currents fairly well( better than arctic waters) and believed that was where the spices were. The use of the Canaries or te Azores would be a big boon, but i don't know enough about the political situation at the time. I've read that England had better relation with Spain at the time, but Columbus had offered the Spanish and Portuguese his mission first, so they countries might be a little ticked off if Columbus wanted to use their islands as a pit stop. Getting from England to the Bahamas was not impossible, but Columbus would have needed a more professional (well paid) crew to stick with him. If he took an arctic route, i would imagine the route would be similar to John Cabot's and Columbus, like in the Caribbean, would only stay long enough to take some slaves and any fancy looking thing he could find. In the short term(2-5 years) it would not really matter because Columbus would lie and say he found Asia either way, but it could determine whether Native Americans would be called Indians or Cathays(northern chinese).
On Columbus' further voyages, he would obviously go south, desperately looking for Natives with gold, which he would have to go to the Caribbean to find. Henry VII was notoriously frugal, so he would probably fund these expeditions by promising Italian explorers land claims, a right to more profits, or titles, which could lure in more talent. Once the English realize this isn't Asia, their further exploration might be far slower than Spain's because the English lacked the crusading zeal of Ferdinand and Isabella, so most explorers would have less support for colonizing, probably searching for a northern or southern passage to asia instead. I think if Columbus found caribbean gold, the English would continue on. If one Englishman saw Tenochtitlan, they would invade until they conquered it. For first colonies though, i don't see a ton of English settlements or Spanish conquests. French style trading posts devoted to selling goods to the natives and enabling further explorers or conquerors are likely.
We have to think about the other European powers too. The Portuguese found Brazil in 1500, and that likely goes the same as OTL. The Spanish were less navally focused than the Portuguese, but they have the Canaries as a base and a surplus of cash from the Reconquista, so they would be soon behind. If the English have claimed the Bahamas by now, Spanish conquest and conversion efforts are more focused on the Northern coast of South America than the Carribean. Since the English wouldn't have intensively settled North America, you could still see French, Portuguese, or other powers setting up colonies there. It is important to note that the First English Colonies had the Irish plantations as a model, but this exploration predates those, so it is unlikely English efforts would resemble OTL British Colonization.
Since i don;t see the discovery of the Americas affecting Henry VIII's genetics, there is a good chance the Protestant Reformation(which did and would have happened independent of American events) still comes to England, so Catholic exodus might drive real settlement colonies. For Conquest, the Spanish Colonies were mostly rules by whoever conquered them until the 1580s when the crown slowly took control. However, Henry VII, Henry VIII, and Elizabeth I all heavily restricted noble powers so they might not be cool with lower class merchant basically establishing military dictatorships across the sea. On the other hand, the English monarchs had much less ability to raise money without parliament then the Spanish monarchs did, so they might grant more independence to conquerors in return for running the Colonies. OTL, the Spanish taking control of their colonies was a huge financial mistake because they didn't pay for themselves until the Bourbon reforms in the 1760s.
It wasn't part of the question, but the English colonizing North America, Central America, and most of the Caribbean would be a disaster for Native Americans because the English were not very interested in converting, were the worst to the natives IRL, and looked down on Interracial marriages more stridently. You can look to their policy of putting down an Irish revolt every hundred years by slaughtering 1/3 of the Islands population for evidence of this.
This would have massive and escalating consequences in Europe. First, England would have the money and necessity to possess the largest navy in Europe, and if they wanted to overtake the Portuguese in the East indies once they finish conquering the great Native American Countries in the 1570s, they could probably do it with ease. You might think this would devastate Spain, but the Spanish were far less dependent on new world wealth, with treasure shipments only ever making up a fifth of Spain;s treasury. Spain funded the largest army in Europe through insanely high taxes that squashed the original thriving manufacturing of Aragon, and then through the massive selling of noble titles, which encouraged the wealthy to spend money on those instead of the economy. All this crippled Spanish wealth. Charles V would probably still happens because his existence was the result of political planning that predated 1492, but his wars were unavoidable. Francis I of France wanted to dominate Italy, no matter the cost, he Ottomans were surging, and the reformation happens. This might lead to a French dominated Europe, which is an entire other really complicated scenario. But otherwise, Charles V reign is hard to tell. If England is much stronger, he might bide his time to marry an English princess, which would lead to a less devout Philip II, which might lead to Spain avoiding all of its devastating wars. The point is, anything beyond the 1550s is really hard to predict in Europe.
I do not think England would have fought a ton of wars with Europe under Elizabeth or James because both reliably avoided foreign war, which Probably leads to a much stronger England. If the Stuart monarchs start a civil war, it might go very well because they have New World wealth to fund armies instead of relying on Nobles, so perhaps and Absolutist England like France?
The main reason why England was so dominant after 1650 was because they copied Dutch capitalism and mercantilism more effectively and used central banking to dominate the world. But, with a POD of 1489, the Dutch might never be formed. The English still have the Hanseatic legacy, but they rise of banking and joint stock companies that allowed Britain to run a profitable empire might not happen. If they do develop a strong capitalism and mercantilism society while also holding most of the new world, then Britain is the world's superpower by the 1600s. Beyond that, it is difficult to tell how the would effect English society long term.
Getting back to the Americas, we probably see slower colonization in non-British America, and maybe an earlier development of slavery. If British conquerors take the Incas and find Bolivian Silver, that region goes to Britain. They would probably conquer Mexico from the Aztecs and take Columbia to connect their colonies. I imagine English searches for a northwest passage, English naval dominance, and their head start would lead England to Dominate North America. Portugal would probably still settle Brazil similarly unless they got into a big war with England, or some other naval power.
Long term, it is extremely difficult to say what would happen to the Americas. Since there aren't many colonies left in the Americas, i would bet these would eventually become independent, but probably in a completely different form to our America. The enlightenment might not even have existed if the circumstances were right. As for Russia, this timeline lead to them not even getting to the Bering strait, and they never colonized it instensilvely.
Overall, it is hard to say how the English discovering America would have effecting the world in the long term, but it is highly unlikely this would have closely resembled Spanish colonization, or OTL British Colonization.