Would there have eventually been an England without Vikings and Norse settlers

Bede, in his Historia Ecclesiastica, posits an Anglalond and a gens Anglorum as early as the 730s. England even has a number of Bretwaldas - 'overkings' - until the 830s, although their power was largely more symbolic than practical.

The capture of the Danelaw certainly catalyses the unification of England, but the process was already, if slowly, underway. With the battle of Ellendun in 825, the balance of power in England begins to tilt inexorably towards Wessex. Alfred's grandfather, Ecgberht, briefly establishes control over Mercia in 830, and secures the submission of Northumbria, but this lasts less than a year. Far more enduring is the alliance acheived by Alfred's father Æthelwulf and the Mercian kings Beorhtric and Burgred. It's tempting to say this alliance was formed in the face of Viking aggression in the 840s, but interestingly Æthelwulf and Beorhtric celebrate the foundation of their alliance by waging a joint campaign in Wales which increases joint 'English' overlordship along the Mercian borders. This alliance was also economic, with Mercian and West Saxon coinage becoming particularly alligned in this period, and with Wessex supplying minting equipment and moneyers to restart the production of Mercian coinage at the London mint in the wake of a particularly destructive Viking raid in 850.

Certainly the Viking Wars of the 870s to the 920s provide the perfect opportunity for the West Saxon and Mercian states to greatly expand their powers, authority, infrastructure and influence, which underpins the rapid coalesence of 'England' in the 920s, as well as allowing both Wessex and Mercia to lay claim to what had previously been Northumbrian and East Anglian territory in the interests of 'liberating' fellow Englishmen, but the process of unification was certainly already underway by the 840s, some 30+ years prior to the establishment of the Danelaw.

As a Danish-held settlement, York was certainly prominent, but it had previously been important as the de facto capital of the Kingdom of Northumbria. Before that, it had been the important site of a Roman legionary headquarters, so it was far from a backwater. That said, the Danish occupation did mean that it grew in prominence as a trading hub, particularly with other Scandinavian settlements on the Isle of Man and Western Ireland.

Would there have eventually been an England without Vikings and Norse settlers?

It's always difficult to speculate when it comes to history, as even the smallest of changes can eventually lead to massively disproportionate effects. However, it is very likely that we would still have seen - eventually - a united England, even without the vikings.

Even before the arrival of the vikings, the idea of the 'English people' was a recognised one. Though the Germanic peoples who had settled in England hailed from several different tribes - the Angles, the Saxons, the Frisians, the Jutes, and so on - they were starting to be recognised as one larger group, even before the 'external threat' of the vikings showed up.

Bede's "The Ecclesiastical History of the English People", one of our main sources on the post-Roman period, was written someone around 720 AD, for example. For reference, the 793 Lindisfarne raid is often used as a neat starting point for viking activity in England, though of course its more complicated than that in reality.

Although we didn't see any direct attempts to unify England into one kingdom before Edward the Elder's reign in the 10th Century, we did see a system of 'overlorship' - simply put, the most powerful of the English kingdoms would demand that the other kingdoms swear fealty to them, in a loose relationship. Until the early 8th Century, this was generally the kingdom of Northumbria, and after that point, generally the kingdom of Mercia. The Anglo-Saxon Chronicle describes the overlord as "Bretwalda", or 'Britain's ruler', though in reality this was more of an English concept than a British one.

We also have the 'Tribal Hidage', from between the 7th and 9th Centuries - we're not quite sure. This lists the amount of 'hides' each English tribe/kingdom contained. A hide was a small administrative unit, with each hide being - roughly - the amount of land needed to support a single household. What's notable about this document is that it includes hidages across pretty much all of what is now considered 'England' - though the border with Scotland was very fluid at the time - despite being compiled before any unified English kingdom came to be.

The existence of the Danelaw likely did speed up the process of English unification: by giving the kings of Wessex a casus belli for conquering the rest of England, by giving the English people a common enemy to unite against, and so on. However, even before the arrival of the vikings, there was a concept of there being an England, of there being an English people, and a tradition of one King holding nominal (if not actual) power over all of England.

Is the prominence of York due to them?

Partially, yes, though York was fairly prominent before the arrival of the vikings. Under the kings of Northumbria, it had been the closest thing to their capital city (though the royal courts of the time would generally travel, rather than staying in one place), and had also been a regional religious centre - one of England's two archbishoprics, the other being Canterbury in the south.

It certainly did flourish under the vikings, however. Though it is of course tempting to see the vikings purely as raiders, they were also accomplished traders, and York became a key part of their trading network. During the time period, the vikings also held Dublin, the largest slave port in Western Europe - York became a useful stopping-point in between Dublin and Scandinavia. A large excavation in York's Coppergate area in the 1970s/80s revealed a highly-developed system of tenements, workshops, and dock infrastructure, all surprisingly well preserved from the viking era. The organisation of these buildings suggests a level of central planning was involved - combine that with evidence of coins minted at York, and we're starting to see clear signs of a well-organised kingdom, centred on York. There's also evidence of goods from across Europe passing through York, including silks from the Near East, and Byzantine coinage.

Sources

For a general introduction to the Anglo-Saxon period, you're going to want to look at Higham & Ryan's The Anglo-Saxon World, which is a fantastic book.

Primary sources such as Bede's Ecclesiastical History, and the Anglo-Saxon Chronicle, are also useful and interesting - though of course, the former is pre-viking, and the latter doesn't look on them too favourably.

For York, the Yorkshire Archaeological Trust and the Jorvik Centre have a fair amount of online resources. Unfortunately, I don't know of many good, accessible works on Jorvik - Aspects of Anglo-Scandinavian York is very thorough and complete, but unfortunately, also very dry,