[Original: Misread the latter part of OP's question about the Danes' adoption of Latin in late Anglo-Saxon England]:
Main problem of OP's original question is that Latin was probably never a means of daily communication for the lay commoners in Late Anglo-Saxon England, so the majority of Dane/ Norse (lay) settler did not have strong incentive to learn Latin in the 10th century northern England.
Matthew Townend, an expert on historical linguistics in Late Anglo-Saxon England, describes England at that time primarily as bilingual society, where inhabitants of either of the Old English/ Old Norse linguistic communities were familiar with the corresponding proper name (like the place name) in another language each other, citing the late 10th century Latin translation of Anglo-Saxon Chronicle (by Aethelwald):
"The body of the ealdorman mentioned above was carried away secretly, and was taken into Mercia, to the place called Northworthy [in Old English], but in the Danish language (Danaam linguam) Deoraby [now Derby] (Townend 2000: 94)."
On the other hand, not so many written evidence, either in Latin or in (Old) French, are transmitted from pre-1066 Normandy. What famous DuDo's History of the Normans and Satire Poem Moriuht by Warner of Rouen testify is the certain degree of appreciation of Latin literature in the court of the duke of Normandy in the second and third decade on the 11th century, but not the wide spread of either Latin or Old French (as a written language) literacy among the duke's Norman followers, I suppose.
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[Added]: Really sorry for misreading the Anglo-Saxon settlers as the Danes in my original post.
I'm not so specialized in Early Medieval British Isles, and the opinions of researchers have been reached an agreement on the linguistic situation of Late Roman Britain. Nevertheless, they have more and more been inclined to identify the increasing numbers of Latin/ Brittonnic (British Celtic) either in Old English or in place names (It is worth checking /u/BRIStoneman's post in: In the early Middle Ages English supplanted Celtic and Latin as England's main language. If you lived in Britain between 450 and 700 what would the transition look like to you?.
To give an example, Oosthuizen argues that these place names with Latin loanword or elements like Portsmouth and Bedford could be the evidence of the survival of Spoken Latin language at least up to the 8th century (Oosthuizen 2019: 43-45). Some of these place names were called interchangeably by individual languages, like Verulamium in Latin and Uerlamacaestir in Old (Early) English about at the same time. This circumstances can be somewhat compared with OE-ON bilingual society in the 10th century England, as I cited above.
References:
(Edited again): fixes typo.