England's first attempts at establishing a colony were under the authority of Queen Elizabeth, "the Virgin Queen", in the 1570s and 80s, although no exploration was done at first, let alone settlement. The first actual exploration in preparation to establish a colony was done in 1584-85. John White, one of the colonists aboard the fleet of ships that sailed to America to explore the land claim and identify a location for settlement, prepared a map which used the name "Virginia" for the first time. The map is actually a coastal part of what's now North Carolina, and is labelled "La Virginea Pars", literally "part of Virginia". A 1591 map of the same area, based on White's map, was the first to use the spelling "Virginia".

The first attempts at a colony, the Roanoke colony, famously failed. In 1606, a new attempt was made under King James I. He granted a charter, split in two, between two joint-stock companies called the Virginia Company of London, and the Virginia Company of Plymouth. In the original charter, the colonies didn't have specific names. The London Company's grant was simply called the "First Colony" and was granted the area south of the Hudson River, and the Plymouth Company's grant was identified as the "Second Colony" and was granted the area north of the Hudson River.

So, the first settlers of the "first colony" of the Virginia Companies set sail the following year. Virginia being the name of England's mainland North American land claim, long named after the now-deceased Elizabeth the Virgin Queen, the new settlement itself was called Jamestown, named after the current King of England.

Once that colony was successful, John Smith was commissioned to explore a suitable site for the second colony, which, according to the 1606 charter, was to be settled at least 100 miles away from Jamestown, and also north of the Hudson River (so, many hundred miles away from Jamestown). Instead of calling it "North Virginia" like they had been or "Virginia II: Electric Boogaloo", it was styled as "New England", a name first used in Smith's 1616 report of his exploration, A Description of New England.

The Plymouth Company petitioned for an updated charter in March 1619-20 under the name "Petition for a Charter of New England by the Northern Company of Adventurers", and according to that charter, it wasn't John Smith who coined the name, but King James I himself (spelling modernization and emphasis mine):

"The most humble petition of your Majesty's counsel for the second colony, and other adventurers in the Western parts of England [i.e. in and around Plymouth, England] for the plantation in the North Parts of Virginia in America...

"Whereas it pleased your Majesty...to give license for the establishing of two Colonies in Virginia in America, the one called the First Colony undertaken by certain noblemen knights and merchants about London; the other called the Second Colony likewise undertaken by certain knights and merchants of the western parts...

"...Your Petitioners do in all humility desire that your Majesty will...First, that the territories where your petitioners make their plantation may be called (as by the Prince His Highness it has been named) New England, that the bounds thereof may be settled from 40 to 45 degrees of Northern latitude..."

The "Second Colony" needed a name, and with the monarch no longer being the "Virgin Queen", and the name "Jamestown" after the current monarch already having been taken, "New England" was appropriate. Whether it was Smith who came up with the name, or King James I or some royal advisor who told Smith to name it as such is lost to the ages, as far as I know. Either way, it was a place in need of a name, and "New England" was suitable. Maybe "Jamesland" or "Kingsland" could have been used, but if considered, "New England" is the one that won out. For all we know, "New England" may have been considered back in Queen Elizabeth's time, but "Virginia" won out instead. Why, exactly, is unknown, but it seems that flattering the Queen might be a good idea if you're petitioning her for a land grant.

There were several settlements in Virginia, all under a succession of charters granted to the London Company, but after mismanagement culminating in the Jamestown Massacre of 1622, the company's charter was revoked, and Virginia became a royal colony, under the direct control of the monarch.

In 1625, James I died, and his son Charles I succeeded to the throne. Colonial charters were granted both to the south and to the north of Virginia, within the boundaries of the original Virginia colony. The one to the south was named "Carolina" in 1629, in honor of the King ("Carolus" being Latin for "Charles"), and the one to the north was named "Maryland" in 1632 in honor of Charles' Queen Consort, Henrietta Maria.

Something similar happened in New England. There were a series of religious schisms among the Puritans, resulting in outcasts from Plymouth founding Rhode Island and Connecticut. They petitioned the crown for their own charters, and they got them. King Charles also granted a charter for a new settlement at Massachusetts Bay, founded under Anglican authority, to combat the local influence of the Puritans. All these various charters were in the confines of the colonial land originally styled as "New England", and the name stuck. The Massachusetts Bay colony was actually founded by the "New England Company", a successor to the Plymouth Company.

This somewhat begs the question: why didn't the area of Maryland, Virginia, and the Carolinas, collectively come to be called "the Virginias" or some such thing? This is harder to answer, but the probable reason is that it would have been confusing, since one of the colonies itself continued on under the name "Virginia". In New England, this didn't happen. The Mayflower Compact had called the area "northern Virginia", and the 1629 charter called Plymouth "New Englande", but then "the Great Charter of New Englande" was surrendered in 1635. Instead, there were separate charters for Massachusetts Bay and for Plymouth, as well as for Connecticut (actually called the "New Haven Colony" in its early charter, but geographically defined by the Connecticut River), for Rhode Island (actually called "Providence Plantations" in its early charters), and for New Hampshire. Without an actual "New England colony" in existence any longer, the collective colonies that had once been within its geographic boundaries could be referred to as "New England".

In Virginia, there still was a colony officially called Virginia, so referring to Maryland and the Carolinas as "the Virginias" might be too confusing. In any case, I don't know of any evidence that it was ever done, and it wouldn't necessarily make sense anyway, since technically speaking, the New England colonies, too, were once part of "the Virginias", since they were defined as "northern Virginia" in the ealiest charters. The absence of calling the Southern colonies "the Virginias" was probably also reinforced by the geographic separation between those colonies, unlike in New England, where the major cities of those colonies are bunched up much closer together.

As an addendum, several decades later, King James II may have re-enforced the collective name "New England", although it didn't really need reinforcing at that point. In 1686, he tried to re-organize all the New England colonies as well as New York and New Jersey under the administration of a new entity called the "Dominion of New England". This didn't last long--the people of Plymouth revolted, and so did the people of New York, both successfully (if temporarily) deposing the government for two years. By the time both rebellions were put down, James II himself had been deposed during the Glorious Revolution and replaced by his daughter and son-in-law William and Mary. William and Mary re-re-organized New England into the more familiar colonies that would last until the Revolutionary War.

TL;DR: England's entire land claim in North America was initially referred to as "Virginia". This became the name of the first colony. Once a second colony was formed, it was styled as "New England", to distinguish the two. The original New England charter was surrendered in 1635, while the separate charters for the northern colonies persisted, or were newly established. The colonies, and later the states, once within the boundaries of the original New England colony continued to be referred to collectively as "New England", and the name has stuck ever since. Virginia and the other original Thirteen Colonies were never within the boundaries of the original colony officially styled as New England that existed from 1620 to 1635.