I would give my first born to the elves if there were a good answer to this question (with apologies to my only child).
First there is a problem with sources: because literacy basically equals Christian, any sources about pre-conversion belief systems are filtered through an imperfect Christian lens. The second problem is that sources are all too rare when it comes to pre-conversion belief systems (or even post-conversion belief systems not related specially to Christianity).
What remains is to attempt to connect far-flung dots - separated by the expanse of geography and also spread over several centuries.
We can do a mind experiment, imagining what an ethnographer going back to the fifth century would encounter in Anglo-Saxon Britain and in other North Sea Germanic cultures - and the picture isn't very helpful. When untethered to written documents and central authorities for dogma, folk beliefs tend to drift a lot. They tend to drift a lot even when tethered. Folklore changes over space and time, so whatever our ethnographer documents in one place is not likely to have been the same in another. And when we add the factor of time, our ethnographer would notice even more drifting.
From this we can surmise that whatever anyone asserts about Anglo-Saxon or other Germanic beliefs systems is managing nothing more than a good guess. We can have vague impressions and we can describe general themes and characteristics for a few players, but the closer we look, the more likely we will be wrong. Everything will remain out of focus.
The answer to the specific question about traditions associated with English and related elves/fairies/etc. is complex, and again not entirely satisfactory. There are mentions of these entities in early medieval literature, but evidence is scarce. I highly recommend this 9-part course by Simon Young, an expert in this field: total time 1 hour 35 minutes. Simon does a fantastic job weaving all of this together - together with all the transitions that have occurred. Again, folklore changes, so whatever the traditions were, we can expect that what was documented during the early modern period was different, even while it was connected.
A great body of evidence is indirect, namely that similar entities were documented in the early modern period throughout North Sea cultures. These social fairies (called by a wide range of local names) seem to be related, and more importantly, they are distinct from those of neighbors farther afield. There is a single tradition that seems to "hang together" from Ireland to Sweden and Brittany to Iceland. There are, of course, local differences, but the fact that these far-flung places have similar traditions suggests that they also have deep roots. It is important that documentation - admittedly sparse - from the medieval period seems to affirm this.
All of this said, can we assert that "goblins and fairies of later English folklore were taken from pagan myths?" The answer is either a definite no or a qualified sidestepping that responds by reframing the question. Premodern fairies were not "taken" from pagan myths. There were likely some beliefs that predate conversion that provided the soil that yielded the traditions documented in Christian literature, but things were by nature in constant flux.
It would be best to say that nineteenth-century documentation of British and other fairies captured evidence of traditions that drifted from time and appear to have deep roots. The further back in time we go, the less likely we will recognize those traditions, and yet common threads might be recognizable to our time-traveling ethnographer who is going back to the fifth century. But for all that ethnographer recognizes, there will be far more that seems surprising and foreign.
This is a more complicated question that you might first think and there are a bunch of different aspects to get into when answering it. I'm going to split this answer into a few different sections for ease of reading, and because these factors are all related but can be treated a little independently.
1. What was Anglo-Saxon Religion like?
We don't know.
2. Okay seriously, what was Anglo-Saxon Religion like?
We don't know much with any certainty. Our surviving sources from this time are not concerned with cataloguing the beliefs and practices of the Anglo-Saxon pagans. Our one, yes ONE, literary source that details the history of the English in this time period, written by the Venerable Bede an English monk in Northumbria, gives very little attention to the pagan practices of his forebears. Bede was living a generation after conversion to Christianity had completely taken England and while he was not a first hand witness to Anglo-Saxon pagan practices they were not that far from living memory, so in theory he could probably tell us a bit more than he actually does, but he didn't care to, because why would he want to record primitive pagan superstition for posterity?
Now he does offer us a few tantalizing glimpses. For example in his explanation on the English names for various months of the year he tells us that for example November was the Blot month, a blot being a harvest/feast/sacrifice. He also tells us for example that the pagan king of East Anglia, Rædwald, kept a temple devoted to numerous pagan deities as well as Christ, (and the famous Sutton Hoo ship burial from the same place and time does have both pagan and Christian elements within it) Bede also claims that the former high priest of the pagan gods in Northumbria personally oversaw the destruction of his old temple. So if we take Bede at face value we can at least posit the existence of some form of clerical structure and physical structures for the worship of pagan gods. However, if they did indeed exist, none have survived to the modern day or been firmly identified.
So even though the names of various English gods and goddesses have survived into modern day, through days of the week most famously, we know next to nothing as to how they were worshiped/venerated, or even if they were. Many people try and fill in the gaps of what we know with parallels to Norse paganism but that is a bad idea for a host of reasons. The tl;dr is that Norse mythology as we know it today is not the religion of the vikings and even contemporary accounts are likely recording a uniquely elite experience with the tradition that was universal in neither time nor place.
We do know, through later Christian sources (and some archaeological sources), that many people believed in local spirits and supernatural creatures, often referred to as elves for example, that were to be warded against and placated through offerings of grain and protective charms/prayers, and while this is an interesting mythological tidbit, it doesn't tell us much about Anglo-Saxon paganism.
One thing we do know for certain was that these pagan communities engaged in ritual sacrifices, both of material goods (often deposited underground or in wells), animal (especially horses), and potentially human (this is quite controversial and is nowhere near a settled issue), however archaeology cannot tell us the why behind the what. We know sacrifices were done, and even know what kinds of animals were common to sacrifice and can theorize as to why, but we cannot be firmly sure.
It is also extremely important to keep in mind that at this time there was likely almost no homogeneity in the religious practices between the various Anglo-Saxon kingdoms. Pagan practices in Northumbria likely bore little resemblance to those practiced in Wessex or in Wight which both probably varied over the centuries. Pagan priests, assuming they existed, likely had different rituals and practices all over the place. The role that the king played in pagan practices likely also varied significantly, and each of the kings in England at the time had little motivation to be exactly their neighbors, or more accurately their rivals. Keep in mind that at this time England was divided into innumerable smaller kingdoms, some of the larger ones survived into Christian times, such as the kingdoms of East Anglia, Wessex, Kent, Northumbria, Essex, Sussex, and Mercia, but scores of other kingdoms (if you want to even call them that) were also around at the same time, and eventually absorbed by these other polities. England was not united at this time, the language was not united, and there is no reason to think that the practices of the English were united either.
3. What was Early Medieval Christianity like?
The Church that the Anglo-Saxons experienced was not terribly dissimilar from the one that eventually converted Scandinavia, though the clerics were mostly coming from Italy and Ireland. The former Roman lands of Italy, Gaul, Britain, and Iberia had all been converted, and the Roman Emperor Charlemagne had started to spread Christianity at the tip of a sword to the Saxons, the Slavs were starting to convert, and the Roman church was starting to take a shape into a more familiar form to modern people.
However there were still some critical differences. Modern practices such as clerical celibacy, private confession, widespread access of communion, and so on were still some time off. However, Christianity had several things going for it at this time that made it stand out among the competing religions and traditions of early Medieval Europe, chief most among these were prestige and infrastructure.
Christianity at this point was the religion of the Roman Empire. The Eastern Roman Empire had been Christian for centuries by this point,, and the Frankish kingdom in well...Francia, which had extensive relations with the English kingdom of Kent, was likewise Christian and under the authority of the Patriarch of Rome, popularly known as the Pope. This association with the most powerful realms in Europe made Christianity appealing as a prestigious good that could be given and traded, as well as influenced. Indeed the first Christian king of England had a Christian Frankish wife who was credited with paving the groundwork for his acceptance of Christianity.
One of the most important aspects of Christianity is of course baptism, and it was a powerful tool in the arsenal of conversion, beyond the influence a Queen could wield (which was still quite considerable). Baptism, and the subsequent creation of God-Father/God-Son relationships was a powerful means of creating cohesion and loyalty in Early Medieval societies, especially given the highly personal nature of rule and loyalty in England at this time. Advanced bureaucracy, coercive legislation, etc.. and all those hallmarks of "civilized" rule were not in operation at this time.
Christianity was also the gateway to greater trade opportunities, centralization, and infrastructure.
Trading was often restricted, or attempted at least, between Christians and non-Christians, and many luxurious trade goods such as wine and Frankish jewelry (popular in pagan Anglo-Saxon England for example) were appealing to non-Christian populations. However of more direct import especially to would be convert kings, were the benefits that Christianity brought to a ruler's administration and efforts to centralize authority. Latin literacy was a pre-requisite for the administration of medieval kingdoms (despite the presence of the vernacular in both Ecclesiastical and Secular literature in places such as England), and Latin literacy came through the Church. Furthermore a king who embraced Christianity could offer a more prestigious religion to his followers (mediated through baptism) that also brought alongside it greater connections, such as trade, to the powerful realms in Western and Southern Europe. This certainly followed in the case of Kent. After adopting Christianity Kent quickly was able to produce innovations such as a written law code and became a local military power.
Finally, even at this early stage, Christianity was a more popular religion, and I mean that in the sense it appealed to the populace at large. As I pointed out above, popular participation in Norse paganism was limited, but this was not necessarily the case for Christianity. While weekly masses in the vernacular were still some ways off for the majority of the population, many parts of Western Europe were more directly engaged in religious practice (and not necessarily in a way that benefited them, I'm sure the peasants who worked on monastic land were not necessarily thrilled to be doing God's work) in a way that pagans in England were not.