I'd like to offer a slightly different view to that of u/MWL1190. It's true that we're never going to find a linguistic census or a complete statistical breakdown of languages in Roman Britain, and indeed that such a thing wouldn't really make sense - language boundaries in the pre-modern world were porous and not always possible even for those traversing them to define. What 'Latin' meant to a farmer in Gaul would be very different to the language spoken by an aristocrat in Rome, or a soldier in Egypt. People could and did shift between different bits of vocabulary, grammar and phrasing depending on where they were and who they were talking to, and it's difficult to draw a line between that kind of dialectical flexibility and what modern people would call multilingualism. This fits the general pattern where questions about the ancient world are usually best phrased qualitatively than quantitatively, because we usually can't be too specific about the latter.

With that said, and acknowledging the many methodological difficulties here (which I'll do my best to set out below), it is possible to get a sense of where and how far Latin became part of Romano-British society, and reason to believe that Latin had quite a high penetration. It’s possible to construct an argument for certain people having a very high level of Latinity indeed.

The 'upper bound' on ideas of British Latinity comes from evidence like this. During his short-lived reign between AD 286 and 293, the usurper Carausius minted a number of coins aimed at promoting his legitimacy in Britain and northern Gaul. These included short phrases, such as Restitutor Britanniae (‘Restorer of Britain’), that he clearly at least hoped would be understood by at least some of his subjects, and more complex phrases, such as the abbreviation RSR INPCDA. In 1998, Guy de la Bedoyere realised that this stood for Redeunt Saturnia Regna, Iam Nova Progenies Caelo Demittitur Alto (‘the reign of Saturn returns, and now a new generation is sent down from heaven above’ – the sixth and seventh lines of Virgil’s Fourth Eclogue [1]. This seems likely to have gone over most people’s heads, but it does show that the showpiece texts of Classical Latin literature were at least vaguely known in late Roman Britain – there must have been at least some expectation that somebody would ‘get it’.

Turning our expectations down from poetic erudition to basic linguistic competence, there are then the numerous writing tablets found in Britain – the most famous set of which are from 'Roman' (usually auxiliary, recruited in what we would now call Germany) soldiers on Hadrian’s Wall, but more recent discoveries have come from the early layers of Roman London (see here). These are mostly low-level business documents, but include schoolroom exercises, which show that you could get an education in Latin in London at the time. Moreover, there are a number of people named in the tablets as having distinctly Celtic names – not necessarily evidence of British birth, as they could have been from the Romanised areas of continental Europe, but a suggestion of the possibility. It does seem that most of these were written by Romans, or at least outsiders to Britain, but then this is also a single context in a single place, in the very early years of the occupation.

More generally, you can look through the corpus of Roman inscriptions from Britain (known as RIB) here – again, it’s difficult to know much about the people who wrote most of these, and a huge proportion come from obviously military sites. However, you do have dedications to local gods and goddesses, such as to Suleviae, to whom a certain Similis, son of Attus, ‘Citizen of the Cantiaci’ dedicated RIB 192 in Colchester. It seems at least reasonable to suggest that many of these dedications to native gods, or to Roman gods qualified with native names via interpretatio Romana (such as RIB 67’s ‘Hercules Saigon’, dedicated by Titus Tammonius Vitalis in Silchester), were made by people raised in the Celtic religion. We do know that Britons took on Roman names – this is a well-attested across the empire, and the most famous example from Britain is the ‘Cataurus ring’ from Fishbourne Palace, inscribed with the name ‘Tiberius Claudius Cataurus’ – clearly that of a Briton, Cataurus, who had made himself a Roman name by adding that of the emperor who conquered Britain in AD 43. So ostensibly-Roman names may mean Romans – or they may not.

There are other sources of anecdotal evidence that show reasons why Britons would want to be able to communicate with their Latin-speaking occupiers. We know that local settlements developed around Roman camps, many times larger than the camps themselves, and that the army depended on local traders to supply it - many of the Vindolanda tablets deal with this at great length. Roman soldiers and veterans often married local women, and that British people took on Roman practices with huge implications for basic features of daily life, such as the food they ate and the way they prepared it [2]. This is a difficult thing to quantify – is the name ‘Julia Verecunda’, recorded for a soldier’s wife on a Vindolanda tombstone, a birth name or an adopted one – like the famous and impeccably Latin ‘Regina’, who died near South Shields but whose grave tells us was born in Palmyra, speaking Aramaic? However, the general process does suggest that Britons took on fundamentally Roman ways of living, and this backs up the Romans’ own sense of what they were doing. In a famous passage from Agricola 21, Tacitus (who was involved in this process elsewhere in the empire) recalls, not entirely happily, that:

‘He [the Roman governor Agricola] likewise provided a liberal education for the sons of the chiefs, and showed such a preference for the natural powers of the Britons over the industry of the Gauls that they who lately disdained the tongue of Rome now coveted its eloquence. Hence, too, a liking sprang up for our style of dress, and the "toga" became fashionable. Step by step they were led to things which dispose to vice, the lounge, the bath, the elegant banquet. All this in their ignorance, they called civilization, when it was but a part of their servitude.’

So Tacitus at least thought that many of the British aristocrats were taking on the Roman language and culture. He was certainly right about the second part, and it’s reasonable to suggest that he was right about the first as well – access to the Latin language allowed you to interact with the state, not least the huge presence (particularly in the north) of the Roman Army, as well as to access the opportunities presented by trade and contact with other parts of the Roman Empire. More broadly, it was a means of attaching oneself to the prestige of the empire, and this was a key motivator behind the Romanisation of elites across the Roman world [3]

The long and short of it is, we don’t know and we have no way of knowing but likely significantly less rather than more.

One way to look at this is as motive, means, and opportunity.

Motive: why would somebody in Roman Britain want to learn Latin? Latin was the administrative and legal language of Britain, as it was in the western provinces of the Empire. It was also associate with the running of the army. Thus, anyone in the province involved in judicial proceedings, administrative matters, or the military would need at least a smattering if not a strong command of Latin. However, the number of native Britons involved in these activities is not clear to us from our sources. Professional lawyers would likely have been few in number. Magistrates and officials were drawn from the ranks of the elite (both from the island and the empire as a whole) and would not have been terribly numerous. The military stationed in Britain, we are told, was mainly drawn not from the local populace, but from other provinces. For others (farmers, miners, merchants, shepherds) there would have been little to no reason to learn the language since it wouldn’t provide a direct benefit to their lives and labors.

Means and opportunity: how would native Britons go about learning Latin? The Romans never seem to have actively pursued a policy of driving out local languages in favor of Latin. As such, there wasn’t infrastructure in place to facilitate mass instruction in the language. Education looked very different and I would direct you to the significant body of literature on education in antiquity, like the work of Rafaella Cribiore and Ed Watts, for a more detailed view, but to suffice it to say here that access to education in Latin would have been almost certainly out of reach for all but the elite families of the province. It’s worth noting that certain groups of people may well have developed a functional use of Latin, such as those who regularly interacted with soldiers, but the emphasis should probably be on the functionality of the language rather than a broader fluency.

Ultimately, we have no evidence for how many Britons spoke Latin. The Romans kept no record of this nor did they try and enforce the language. However, given the significant barriers to language acquisition (especially financial and pragmatic ones) it would perhaps be safe to say that few Britons learned the language and very few to a level beyond the functional Latin needed for day to day interactions.