It is known that Britain's history of invasion goes as: Celtic arrival, Roman domination, Saxon settlement, Nordic settlement, Norman invasion. If England's identity was largely made from the Saxons (eventually Anglo-Saxons), and there are almost no Celtic words left besides proper nouns, how come there is a significant number of English words that came from Latin, even after excluding borrowings and loanwords from French, if the Saxons and Vikings supposedly overrode any culture there?
During the Middle Ages, Latin was the language of scholarship throughout Europe. Many Medieval Latin words were imported directly into English, either to express concepts that had no exact English equivalent, or to show that the writer/speaker knew Latin and was therefore a learned scholar and presumably authoritative.
As jk says, there are very few Latin loans in English from pre-Saxon times. English does have quite a lot of words borrowed from Latin and Romance, but the vast majority of them come from well after the Saxon invasion.
It's worth noting also that there was a lot of contact between Latin/Romance and Germanic all throughout Europe. When we see Latin words attested from Anglo-Saxon times, it's likely that they were borrowed into Saxon before the invasion, rather than persisting from the Latin spoken by Roman colonists.
Some Latin loans in Anglo-Saxon are really old borrowings that were acquired by the West Germanic speakers from the Romans on the continent (fenester "window", modern German Fenster belongs into this category), other were introduced by Christian missionaries. Since anything Christian was foreign to the Anglo-Saxons they used a lot of borrowed words for religious terminology. I have also heard (but no hard reference on this) that the Christian missionaries deliberately replaced a lot of the heathen religious vocabulary with foreign terms in order to cut the old traditions off.
I'd be very interested in hearing of any words except place names that were borrowed from British Vulgar Latin.
Because that's called Gaelic.
The Celtic cultures, and (to a lesser extent) the Celtic people, were largely pushed up into Scotland or over into Ireland. The new settlers who displaced them were Latin-speaking Romans, then Germanic-speaking Angles, Saxons and Vikings, followed by French-speaking Normans.
For the language to adopt significant numbers of words from Celtic, there would have needed to be an overlap between the cultures (e.g. like the French-speaking Normans ruling over the Germanic-speaking Anglo-Saxons), rather than a displacement.
Incidentally, this tangentally relates to why many 'processed' foods come from French (e.g. "Beef", from "bov", meaning "cow"/"ox"/"cattle") - the language spoken by the Nobles, either because they were newly arrived or attempting to ingratiate themselves with the new leadership - while the name of the source plant or animal comes has German roots (e.g. "Cow", from "Kuh") - the language spoken by the serfs and villein. In other words: did you work to raise it in one form, or afford to eat it in another?