I think its important to note Latin was the official language, but Brittonic would have been the everyday language of the everyday people.

British Latin was largest spoken in the south east of modern England amongst the elite of society. Brittonic was spoken amongst ordinary people. British Latin being based mostly in the South East is also where in time, the area that saw the bulk of Anglo Saxon settlement and the introduction of the Germanic language.

Some Latin survived into modern Welsh, along with Cornish, the modern Brythonic tounge. Words such as Pont, which mean bridge, survive to this day as a legacy of the Romans. Words such as aur (gold) and eglwys (ecclesia-church). In this case, similiar to modern French, Eglwys-église. Llyfr-Livre. And so on. The Roman influence is there to see today!

With Roman influence gone and a heavy Saxon influence in the South East, its easy to see how the language was replaced with such a heavy Germanic influence. One must also consider how long it takes for the Romanization of peoples, as language is a good example, is far much longer than a conquest. It could take generations and generations for British Latin to have been normal for everyone.

Far fetched Britannia held few attractions for Romans too! Latinization was confined to cities and military bases, afterall it is difficult to force languages onto peoples, people will always speak what is easiest for them. For everyday Britons that sees little space for British Latin, which is now competing between Germanic and standard Britonnic. With a large Brittonic influence, one must consider how Brittonnic terms would have infused with British Latin, perhaps making it a less purposeful language in everyday life.

With this considered, Rome had less of an impact culturally the further west and north you go in Britannia and this is why the Germanic settlements of the heaviest bastion of Vulgar Latin speakers in the South East is so important. 'Real' Latin survives all the same through the British church!... at least for those educated religous folk, as will be the case for a long time afterwards! Britain was cut off from the continent, the Frankish Empire would of course become the Holy Roman Empire in its first form, that is important for the continuation of Roman tradition and language. The same cant be said for the wet isles. (Edit. Refer to the comments for more info on the Franks and how their higher connections and respect for rome played a part in the upholdings and adopting of roman culture.)

The Saxons had their own culture and language, they did not need the Roman one. In short, they had no need for adopting another cultures culture xd They were doing pretty well on their own. But with their conversion to Christianity, they'll soon learn to appreciate Roman works and teachings xd (edit: Refer to BigNordDaddy's reply below on this matter. Ive been quite vague and lazy here, but below BigNordDaddy has addressed this point beautifully.)

In truth, its such a difficult question for historians to address because we really cant be sure about definitite awnsers! It is said to have died out in its final pockets around 700 AD, Welsh and Cornish surviving on as the main language of the Britons. Welsh remained a majority langauge in Wales until the turn of the 20th century, and is now trying to survive against the modern equviliant of the Saxon tongue. You just cant escape English xd Maybe there are parallels you can draw from the modern Welsh and Cornish experience. How large groups of settlers in particular can effect local language groups.

I hope this helps. Feel free to add to my anwser! Knowledge is power!

I can attatch a Bibliography if youd like.

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With such a long period of Roman rule, why didn't the people in Britain speak a Romance language? Why didn't the Anglo-Saxons adopt the culture of the Romans as the Franks did?

The British isles experienced marked demographically changes following the separation from Rome at the beginning of the 5th century. The economy declined and became more locally orientated, with cities decreasing in size and population, occasionally even being abandoned outright. Food production suffered as a result of a Little Ice Age in Late Antiquity and in the South of England, a significant portion of the native Brittonic speakers (likely an elite of sorts) left Britain for the European mainland during this period. These Britons settled in what is now Brittany (hence the name) and their language lives on, in France, in the form of Breton. In addition to all this, the so-called Plague of Justinian (probably a strain of Bubonic Plague) also arrived in Britain during the half of the 6th century as well, killing large numbers of people. Exact figures for England are impossible to give, but this pandemic is often estimated to have killed between 20 to 50% of Europe's population as a whole and would have hit the more densely populated cities far harder than the countryside.

All in all, 5th and 6th century Britain was noticeably less densely populated, less urbanized, less organised and more fragmented than it had been some two centuries earlier while still under Roman rule. Generally speaking, when discussing Post-Migration Period Europe, a pattern tends to emerge. In which the more densely populated and more organised areas (i.e. with relatively much of the urban infrastructure remaining) tend to assimilate the languages of the Germanic invaders.

France / Gaul makes for an interesting comparison:

• Northern Gaul (Belgium and the Southern Netherlands) found itself into a similar situation to sub-Roman Britain: it had few still functioning cities (and those that still existed, were small) in addition to a low(er) population figure owing to various past conflicts and invasions in the area. Today, this area is Dutch-speaking, a Germanic language.
• Southern Gaul survived the Migration Period relatively unscathed. Its cities, while decreasing in size, had largely survived and had retained their elites as well as ecclesial infrastructure. It remained Romance-speaking; even retaining a dialect (Occitan) which shows significantly less loanwords from Germanic languages than French (proper) does.
• Central Gaul (between the Somme and the Loire) was kind of in between. The Franks settled the area north of the Seine River and their elites controlled the land as a whole, but in the end; after some 4 centuries, they assimilated. Most likely after already having been bilingual for quite some time. Today this part of France is, with the exception of Brittany, Romance-speaking.

The exact process of language shift in England is not completely known, nor is the extent to which Romano-British was spoken in Britain prior to Germanic settlement. Regardless of how widely spoken or understood British Romance was, it does appear that the British countryside retained its pre-Roman (Celtic/Brittonic) languages to a higher degree than that of Hispania or Gaul. Given that at the very most 25-30% of Roman society lived in or near a city, the Celtic-speakers would have likely outnumbered the total number of British Romance speakers, especially when considering that the degree of urbanization in a far away corner of the empire would have been lower than the aforementioned average, which also includes Italy, Greece and the Eastern Provinces.

Modern scholars tend to explain the language shift from Celtic/Romance to Germanic as follows: Brittonic / Romance -speakers came under the influence of small Germanic warrior elites; whose dialects (being the ones with the most political power) were considered more prestigious, promoting bilingualism. In the case of British Romance-speakers, they might have already been in the process of shifting (back) to Celtic dialects at this point. Political power seems to have been the primary determinant of the superior prestige of Germanic dialects, as the native Britons do not seem to have had much of a technological or organisational edge over the Germanic settlers. The fragmented nature of the post-Roman fiefdoms strengthened the position of Germanic dialects in Southeastern England from which they continually expanded.

Over time, the less prestigious Celtic and/or Romance languages would cease to be spoken. This hypothesis also assumes, that the first to become bilingual would have been the remaining British elites, due to them having most to gain from political and economic ties to the Germanic warlords. Acting as intermediaries, these people would have been the true facilitators of the spread of the various Germanic dialects.

Genetics seems to support this (as there seems to be no genetic evidence for a Celtic genocide in England) as do the etymologies of the names of various early 'Anglo-Saxon' rulers, which are clearly Celtic in origin. Language shifts like this can happen over the course of a few generations and could have been caused by a surprisingly small group of Germanic speakers.

Some small side-notes to your question:

Even Romania, which had been under Roman rule from 106 AD - 275 AD, only 169 years spoke a Romance language.

It is unclear if Romania remained Romance-speaking following the Roman withdrawal from the area. There are strong archaeological and linguistic indicators, that Dacia / Romania was initially depopulated and later resettled by groups of Romance-speakers not native to Dacia, but originating from the Balkans.

TLDR:

Subroman Britain was already politically and economically weakened when the first Germanic settlers arrived. The Anglo-Saxons formed a warrior elite, whose language was more prestigious; encouraging bilingualism and eventually monolingualism among the native Britons.

Sources / Additional reading:

• Linguarum Diversitas: Latin and the Vernaculars in Early Medieval Britain, by D. MacManus.
• The Anglo-Saxon World, by N. Higham and M. Ryan.
• Why did the Anglo-Saxons not become more British?, by B. Perkins