At the core of your question is a problematic term that often leads to misconceptions about historical forms of folklore - namely that word "mythology." It is often misused, and it can have different meanings for different people. In general, it is probably best not to use the term for the folklore of contemporary or near contemporary people, because referring to other's peoples beliefs and stories as "mythology" can be derogatory: we can speak of the Resurrection story, but if we refer to the Resurrection mythology, an entirely different meaning emerges that is disparaging to the sincere beliefs of living people.

So let's figure out what we really want: I assume what you're after is a collection of pre-conversion narratives that give voice to pre-conversion folklore in a way similar to the collections of stories from Egypt, Greece, and Scandinavia that you mention. The simple answer is that this collection does not exist for the English people. A complex answer is that this sort of collection does not exist for Egypt, Greece, and Scandinavia - at least not without a lot of caveats and not the way these "mythologies" are popularly perceived. So let's consider those collections to figure out what the English do and do not have.

Ethnographers and folklorists working on every continent consistently realize that there is no set, "definitive" form of a folk narrative or belief. Folklore changes - from person to person, village to village, and moment to moment. The reason why a given story is regarded as "definitive" is usually because it is the only one that was published and made widely available. Diving deeper into archives, one can often find different, sometimes contradictory forms of that narrative.

With that in mind, we can project backwards to those who wrote their respective "mythologies." Now, we may hold these myths dear because they are great stories and they provide glimpses into the beliefs of former times, but we must realize that these are imperfect windows into the past. The Greek and Egyptian authors were usually writing at a time when belief was still viable; they likely wrote stories that could still be heard, but we should never delude ourselves to think that these were anything more than single frames cut from larger films. In other words, these authors could have just as easily recorded different, sometimes contradictory versions. In addition, since ancient Greece and ancient Egypt spanned many centuries, and since we know that folklore changes over time - sometimes even more than over space - we can expect that there were all sorts of wildly different narratives that could have been recorded.

The situation in Scandinavia is much more complex since the stories that appear in primary sources were written by authors well after conversion. The stories that we have suffer from the same problems I describe for Greece and Egypt, but we must add to that the effect of conversion and of access to larger bodies of literature, everything from the Bible to Classical mythology itself. All of this had an effect on what was recorded, and the effect is compounded by the fact that the context and belief and narrative had changed dramatically because of conversion (even if one argues that conversion was often superficial in those early centuries).

So, returning to the English problem: clearly, we do not have anything on a scale of what was available in Greece and Egypt, or even in Scandinavia. The few early sources we have are wonderful stories, but they are affected by conversion and literacy in the same ways as the Scandinavia sources and the English ones are less plentiful.

The thing we do have in England is a fairly good body of collected folklore during the great age of folklore collecting in the nineteenth century (give or take a decade or two). This material doesn't stack up the way enthusiasts of mythology would hope: counterparts of the Scandinavian gods are gone, for example. But we can just as easily argue that the stories of those Scandinavian gods offer only limited information that requires considerable source criticism.

I find the insights of Swedish folklorist Bo Almqvist (1931-2013) of particularly use here. He discusses a common inheritance from Ireland to Sweden and Brittany to Iceland, belief and stories that share a great deal with regard to the supernatural beings known variously as elves, fairies, hidden folk, pixies, and various other terms. Almqvist points out that his elder Norwegian colleague, Reidar Th. Christiansen (1886-1971) also noted these similarities and the two point to a common inheritance of North Sea folklore. These may not be "gods" - but what are gods, after all, but supernatural beings with a great deal of nevertheless limited power? These shared supernatural beings point to just the sort of "inheritance" that Tolkien yearned for, and one need only look to the vast folklore collections of Britain to find that material.

That's not an entirely satisfactory answer. I understand what you're after: I learned to read Beowulf in the original Old English because long ago in a previous century, I hoped to find vestiges of that early mythology. What I found was a truth that is more nuanced, and while is not as immediately satisfying, I have found it enlightening beyond my expectations.