I'm going to touch on the original Germanic invasion of the British Isles and its effects upon the population's Christianity before I move onto the Vikings. I think it is important to understand how these two phenomena, while so similar on the surface, were not quite identical and resulted in two quite different outcomes.

Part 1

Christianity in the British Isles had indeed undergone quite a sharp decline in the years after the collapse of Roman authority in the area, and almost certainly a contributing factor in this was the influx of the heathen Germanic peoples, according to the Venerable Bede, the Angles, Saxons, and Jutes. Even if Christianity did not disappear from the Britain entirely, it was undoubtedly not as prominent as it was under Roman rule. Why didn't this repeat itself when a new group of Germanic heathens started showing up off of the isles of Britain?

Britain was hit hard during the fall of the Western Roman Empire, and not only did they have to contend with Saxons and their continental brethren, there were the ever present groups such as the Picts and Irish who were also quite keen to make trouble for the British. At the same time Britain experienced a veritable economic collapse. Indeed even relatively basic features of Roman life such as wheel spun pottery disappear from the record. Cities were abandoned, old Roman forts became centers for warlords and their followers to hold up, ad the British spent as much time fighting each other as they did their rivals, both native to the Isles of Britain, and from the continent.

Roman public life and the much discussed "villa system" of Roman Britain, debatably already in a state of decline by the 5th century, ceased to exist almost overnight. Some vestiges of Roman identity were maintained by the populace over the course of the following centuries, and among this was of course Christianity. However without the apparatus of the Roman state and the survival of large scale urban areas, Christianity seems to have withered, and eventually in much of the Island been displaced by the new comers.

However the Christians who still lived in Britain and found themselves under Anglo-Saxon (I'm just going to short hand to this) rule were not wiped out wholesale, their communities destroyed, and identity forcefully erased. Instead, the archaeological picture is actually quite muddled on this. The old narrative of all England being scoured by the newcomers and subsequently free of British influence is totally misguided. British communities and people lived on under these pagan kingdoms. However many historians contend that over years of rule by Anglo-Saxons British identity, and Christianity was a critical part of this, was discarded by the native population in favor of identity as Anglo-Saxons. As a part of this cultural change Christianity was abandoned in favor of the heathenism of the new comers.

Now historians disagree over why this happened. Robin Fleming in Britain After Rome argues that this was essentially a peaceful and natural process of the new found presence of Anglo-Saxons in Britain. She argues that British and Anglo-Saxon communities more or less coalesced with each other and that there were no firm and immutable differences between the two cultures. After the Anglo-Saxons showed up, they and the British basically did as neighbors do and intermarried and formed new communities that.

Guy Halsall in Worlds of Arthur argues that this was process was undertaken by Britons in order to align themselves with the Anglo-Saxon arrivals, more of a "if you can't beat 'em, join 'em" approach. The major difference between the two is that Halsall still adheres to a more violent view of Anglo-Saxon arrival in Britain, even if not the old genocidal view of the 19th century.

Peter Heather in Empire and Barbarians argues that British cultural identity was already quite malleable (as it was across the empire) and that the arrival of the Anglo-Saxons merely tipped it in one direction, whereas in Gaul and other parts of the Empire, the malleable cultural make up of these migratory groups was eventually subsumed into Roman styling and culture.

Regardless of which view you hew more towards, Christianity was by one process or another edged out as the dominant religion among the Anglo-Saxons.

Part 2

So how is this different from the Viking attacks?

Some tempting answers are unfortunately pretty easily dismissed. Perhaps more Anglo-Saxons migrated to England than Vikings did a few centuries later. With fewer numbers their impact on the native Christians would be lesser naturally. That would explain the relative lack of impact of heathenry onto England. However the trouble would first be getting any historian or archaeologist to come up with an accepted number of migrants in either case. The numbers of migrants supposed to have arrived has been in a state of flux for decades. In the 19 and early 20th century it was commonly held that whole Germanic societies uprooted and traveled to England slaughtering any Britons they came across. This view is no longer held to be blunt.

The viking conquests were no different. Although known for brutality and martial ability, the vikings did not wade ashore at Lindisfarne keen on ethnic cleansing, nor was that their goal with the arrival of the great heathen army nearly a century later. While the number of Scandinavians who arrived cannot be established any more firmly than the numbers of Anglo-Saxons, they did not come ashore and wholly replace the now Christian population. They undoubtedly remained a minority and their heathen beliefs very much a minority in Christian England.

At the same time many vikings seems to have taken to Christianity fairly rapidly. Sometimes of their own volition and sometimes not. The viking warlord Guthrum was famously baptized as a part of his conditions of peace with Alfred the Great, and it is not hard to imagine other vikings converting as a condition of peace or to ingratiate themselves with native Anglo-Saxon communities. This process would only have accelerated in the conquest of the Danelaw by Wessex.

Paradoxically, like the Britons several centuries earlier, the Norsemen had little to gain by holding onto their religion and quite a lot to gain by conversion. Political legitimacy, peace, and pressure from powerful Christian realms all incentivized conversion to Christianity.

And this ties back into the different context of the viking invasions from the earlier Anglo-Saxon migration. When the Anglo-Saxons came to England they was no one unitary British state that could oppose them, legends of King Arthur notwithstanding. The economic and political collapse brought about by Roman collapse prevented that. However the viking invasions were not taking place in the same context. Several strong Anglo-Saxon kingdoms were established by the time the norsemen arrived. Some of these were indeed subsumed, Northumbria, East Anglia, and parts of Merica, but Wessex (and its take over of Mercia) withstood them.

Even in the conquered areas though Christianity survived. The presence of a vibrant (and indeed resurgent) English Church in Wessex surely was a part of this. However even in the subjugated areas, Christianity quickly re-asserted itself. The reason for this I believe goes back to the advantages of Christian identity over heathen identity.

Now I do not want to give the wrong impression. Britons and Vikings alike did not all sit down one day and think to themselves 'well my old religion is now politically disadvantageous, better give it up and go with the new one'. That just isn't how these sorts of processes work, but over time the old belief system was just worn down by attrition. This process could take decades to fully be realized, and undoubtedly it was not simply a matter of one day just putting up a statue of Wotan or discarding one. Nor did Wessex eradicate the viking heathens that remained in England through their conquest of the Danelaw.

tl;dr

No one cause or event can be singled out as the defining reason for religious change among the vikings or the Britons before them. It was instead a combination of factors, but ultimately the deck was stacked against maintaining heathen practices in England. Native English Church power was too strong, the remaining Christian polities discouraged it, and there were some severe disadvantages by maintaining the old Norse religion in the face of Christianity.

Sources

The two works that are most important for this answer that I've constructed are Fleming's Britain after Rome and Halsall's Worlds of Arthur. I've also mentioned Peter Heather's Empire and Barbarians.

Besides these there are a few other sources that I've read and would be worth a look if you're interested. Harting's The Coming of Christianity to Anglo-Saxon England and Gwyn Jones's A History of the Vikings are both pretty important for this.

For primary sources, I mentioned Bede and Gildas each. Bede's Ecclesiastical History of the English and Gildas's sermon "On the Conquest and Ruin of Britain" to be specific. The Anglo-Saxon Chronicle, Gregory the Great's correspondence, and the Norse sagas (and Saxon Grammaticus's Deeds of the Danes) are also worth a look as well.