Old English played the role of commonly written language much earlier than Old Norse. (There are vanishingly few manuscripts/texts from that time!) The Nordic literary tradition (in Latin too) only truly fired up after Christianization, besides runic inscriptions (which are rather short) there wasn’t much until ~1200 AD when e.g. the Icelandic sagas started popping up (though they were most likely already oral) - since the saga age they besing was ~1000 AD.
More importantly, OE and ON are extremely similar. Having studied OE (and knowing other Germanic languages), I understood ~80% of ON right off the bat. There are many books like Graeme Davis' Comparative Syntax of OE and Old Icelandic which go into depth about this and just name them (and to a lesser extant Old High German/Continental Germanic languages) as fully comprehensible parts of a dialect continuum. After a week learning low hanging fruit of noncognate words and a few key differences (namely the middle voice) I felt equally proficient in them. (In speech, this would be a bit worse as their pronunciations didn’t coincide, but it’s nothing people don’t get used to quickly as e.g. Portuguese' w i d e pronunciation shows.) (There are also minor issues like different genders in some cognates, which some posit led to the English declension system’s collapse, especially combined with vowel reduction in unstressed vowels. The Swedish vowel shift is called the Great Vowel Dance.)
Regarding Christianization, David Wilson’s paper The Viking Relationship with Christianity in Northern England argues they quickly added the Abrahamic God to their pantheon and respected Christian burial customs. (There’s also work on the Christianization of Scandinavia and elsewhere re: Jesus as a war god at first, which seems analogous.)
Some of the diocese of York’s bishops (Oscytel, Oswald) were Danes, indicating that many were indeed educated and Christianized since Guthrum’s conversion. Oda, the (was it already arch- at that time?) bishop of Canterbury in the early 900s was even a Dane (and Oswald’s uncle)! Unfortunately his chronicler only wrote in OE.
In spite of these Scandinavian figures, little ON was written in England. Thordr Kolbeinsson wrote some stuff during Cnut’s lifetime about him in ON (his patron was involved in taking the English throne). I’m not personally aware of anything earlier than that of significant length - although there definitely are some long runic texts. Now, the crux’s that Thordr didn’t write in England. People sure wrote about Cnut in England in OE, the ON texts are from Scandinavia (and a few centuries later.)
Interestingly, in Scandinavia Runes disappeared quickly after Christianization, but remained in England (in OE) until the Norman conquest (St. Cuthbert’s coffin in 700 AD all the way to the Beagnoth sword, besides various rings etc.) Perhaps due to this, there don’t seem to be many Norse runic inscriptions in England - but they are still there, mostly Christian ones (even on crosses), especially on the isle of Mann. In Edinburgh there stands a stone at Princes Street Garden which is thought to have been carved by the same Erik who made more in Sweden.
Unfortunately, there just aren’t any long texts in Latin letters from this time in ON (I think Thordr wrote in runes, which were later written down) although we had very educated men who must have spoken it, writing in other languages. Snorri says Thorgilsson was the first historian in ON which would have been ~1100 AD. I couldn’t find any Norse rune stones in England proper - no monks, students or anything carving them, sadly.
N.b. This was an educated world. Scribes and Scholars details 2 Greek missionaries in England around 700 AD, who Bede relates taught Greek at Canterbury.
On Secondary Sources
This guy discusses an excerpt where a Norse poet composes Norse poetry for an English king and the writer states that they are 1 tongue: https://youtu.be/BaWgJq9OVGM?t=521
Mathew Townend’s Linguistic Relations between Speakers of Old Norse and Old English is the only direct work. Unfortunately the majority of ON texts come from Iceland besinging their adventures… Everywhere. So there’s not the same kind of evidence like ancient authors directly discussing the interaction of Greek and Latin.
The Oxford Handbook of the History of English around page 500 has some pertinent sections and covers theories on modern English as a Creole of ON and OE (although the Anglicization of Celts seems like a better cause - Townend goes into this a bit) and even contrasts Old French as a prestige language and ON as an “adstratal” language of equal prestige to English.
Sara Ponz-Satz' Norse-Derived Vocabulary in late Old English Texts may hold insights but the format is rather… strange, ultimately focusing on each individual loan word used by a bishop.