I agree with at least most of J. Siebeneichler's & Sempaiscuba's answers and add the following.
It is true that as much of Western Europe Latin continued in use in the British Isles after the end of the Western Roman Empire within the Church, and hence for ‘book learning’ generally, which was for a time mostly a church thing.
However, the Geoffrey Thomas's original question is right that as a language of everyday life and as people's first language Latin quickly and disappeared in Britain, unlike in almost all other ex-provinces of the Western Roman Empire, and that this is important and requires explanation.
The only Pre-Roman languages that survive in the former Western Empire are Welsh (and its ancient offshoot Breton) and Basque. (The East, over much of which Greek was the main language of government and trade, has a different linguistic history)
In the mainly lowland parts of south and east Britain that were conquered by the Anglo-Saxons and became England, Germanic dialects prevailed, and developed into the English language.
In the mainly upland parts of the west where the native peoples initially avoided conquest, the pre-Roman native Celtic language Welsh (initially just called 'British') predominated.
The question may therefore be divided into:
Why did the Anglo-Saxon incomers not adopt a local dialect of late Latin as e.g. the Franks eventually did in Gaul, the Lombards in Italy or the descendants of the Visigoths and Sueves in Spain and Portugal? (Such dialects gradually becoming the newly differentiated but almost all Latin-based languages such as French, Languedoc, Italian, Castilian and Portuguese.)
Why did the Celtic language speaking native population of former Roman Britain either remain Celtic (Welsh) speaking or else adopt English? There was no equivalent to e.g. formerly Celtic-speaking Gauls adopting the late Latin dialect that became French.
We do not know the details of how this happened as the key period c 400 - 600 AD is one of the most obscure in British history, from which almost no written records survive, apart from one short book/ sermon 'On the Ruin of Britain' by a monk called Gildas. Attempts to reconstruct the history of the formative period of England therefore rely heavily on works written centuries later, or the rare references to Britain by Continental European writers of the time.
It is possibly significant that:
-Britain was only permanently occupied by the Romans from 43 AD, a century or more later than e.g. Gaul, Spain or northern Italy. Roman rule in Britain seems to have ended abruptly around 410 AD, earlier than most of the rest of the Western Empire. Roman civilization and Latin language thus had less time to become generally adopted.
While it is hard to prove or disprove, I have heard an archaeologist seriously suggest (in a talk to a Historical Association) that the effects of being 'last in, first out' of the Roman Empire are felt even today in Britain’s ambivalence about being 'European'.
However, this argument must be qualified. Britain was under Roman rule for more than 400 years, quite a long time; about twice as long as the United States has so far existed and more than twice as long as, say, British rule lasted in India.
-Possibly a more important reason for the very early disappearance of Latin as an everyday spoken language in Britain, unlike in e.g. France or Spain, was that the Anglo-Saxon influx into Britain differed from the Germanic influxes into France and Spain in that the Anglo-Saxons were still pagan and largely illiterate (despite some short Runic inscriptions). They had had less contact with the Roman Empire than say the Franks and Visigoths by the time those peoples conquered France and Spain (both Franks and Goths had by then already adopted a form of Christianity). Accordingly it was easier and more inviting for the Franks and Visigoths, already somewhat more Roman influenced and having more understanding of how Roman government and society had worked, partly to preserve and slowly to merge into the Latin-speaking societies they had conquered. The pagan Anglo-Saxons were more likely just to destroy and replace.
-Also, the south and east of Britain, closest to the rest of the Roman Empire and, to judge by the number of Roman period towns and villas were by far the most Romanised parts of Britain, and hence the parts most likely to have adopted Latin speech.
Further north and west, places like the hills and mountains of what are now Wales, Cumbria and Cornwall show less sign of Romanisation, apart from having more Roman army forts, presumably to keep the still half-barbarian inhabitants in subjection, as well as to keep out even wilder barbarians from what are now Scotland and Ireland. Consequently, although individual Latin words were absorbed into Welsh, native British Celtic speech probably persisted more strongly in these areas.
It was the misfortune of any Latin-speaking Romano-Britons there may have been in the South and East of the island that their regions, having the richest farmland and being closet to continental Europe, were the ones most quickly and thoroughly conquered by the incoming Anglo-Saxons, and hence were the most complete language replacement occurred.
-I have also read what I think was only a half-joking suggestion by a professional historian, I cannot remember where unfortunately, that following the Anglo-Saxon Conquest most of Britain became purely English speaking because 'the English have never been good at learning foreign languages'!
