Really now? I'd like to see some evidence for that.
Many of the latinate words either come from the influence of the catholic church, or entered via english loanwords, not from the romans themselves.
On the contrary, loanwords from Latin penetrate even the most basic strata of the language and can be easily dated to before the rise of the Catholic Church or Anglo-Saxon arrival in Britain. If we leave aside "cultural" words for which there would have been no native Brythonic equivalent (book, letter, school etc), there's still a remarkable number of Latin loanwords which have become the basic or only word for the concept in the Brythonic languages. For example:
coes 'leg' < L. coxa
braich 'arm' < L. bracchium
asgell 'wing' < L. ascella
barf 'beard' < L. barba
boch 'cheek' < L. bucca
corff 'body' < L. corpus
corn 'horn' < L. cornu
gwain 'vagina' < L. vagina
And that's just in the realm of body parts. For comparison, in English a Latin- or French-derived term for a part of the body is rarely the basic word, and there's always a solid Germanic synonym which remains in use (e.g.
stomach vs
belly): in Welsh these are the basic or only terms and the original Brythonic words have not survived.
As it happens, the Latin element in the Welsh lexicon is the part which has received the most scholarly attention. Falileyev's 2002 article
Latin Loanwords in Old Welsh is comprehensive and up to date, although deals only with the rather limited corpus of Old Welsh (it is telling, however, that even in such a small number of documents the evidence for a thorough Latin lexifical penetration into the language is overwhelming). Haarmann's 1970 work
Der lateinische Lehnwortschatz im Kymrischen is an excellent statement of the current common opinion in Celtic linguistics on the Latin influence on the Brythonic lexicon: the essential summary of the evidence is that Brythonic experienced a massive influx of Latin loanwords during the period of the Roman occupation, which were often mediated early by the Roman army rather than later by the Catholic church. I can supply the evidence for why it's thought that the army contributed more than the church if you like, but this post is getting rather long as it is without even addressing the Latin structural influence and its implications. However, before moving on I think it's worth noting that the native term for the autochthonous inhabitants of Britain in all three Brythonic languages is in fact a loan from Latin rather than a continuation of the Celtic original.
Schrijver, in his 2002 article
The Rise and Fall of British Latin makes the point that "Lexical borrowing represents the most superficial level on which languages can influence one another. Particularly in the case of Latin, influences on a more structural level are to be expected." And this indeed is what we see. Let's look at phonology first, and then morphosyntax. Schrijver lists six phonological areas in which Latin influenced Brythonic, I'll limit myself to the three which to me seem the most significant:
Proto-Celtic, as far as we can determine, had word-initial stress, a feature which still obtains in the Goidelic languages. In Brythonic this was retained until about the second century CE, before shifting to a Latinate penultimate stress. This is significant for two reasons: it allows us to reliably date a number of Latin loanwords to before the influence of Christianity; and it is during this period that the Vulgar Latin of Gaul underwent a series of syncopes and apocopes to arrive at a similar system (we can therefore reasonably assume the same for the spoken Latin of Britain). Again, it is significant that from our evidence Gaulish underwent the same changes during this period.
In spoken Romance during the first few centuries CE, we see a loss of final nasals in polysyllables but not in monosyllables. We see exactly the same phenomenon in early Brythonic. Thus we find nasal mutation in Brythonic triggered only by monosyllables (e.g.
fy nhŷ 'my house' < *
men tegos) rather than the situation that obtains in Old Irish, where nasal mutation is triggered by any word ending in a nasal consonant, regardless of syllable structure: compare Welsh
y pen bach without nasalisation with Old Irish
a gcenn mbecc with nasalisation, both from a putative Proto-Celtic *
sosin kʷennon bekkon. Given the importance of mutation in the later Celtic languages, this is not a trivial change.
Proto-Celtic, like Classical Latin, had a vowel system which opposed long vowels to short vowels, a situation which persisted into Old Irish. Brythonic and early Romance, however, collapsed this quantity-based system into one primarily based on quality. The manner in which this took place is tellingly similar to what happened in the Romance of northern Gaul. We can assume that both Latin and Brythonic had a vowel system
ī i ē e ā a ō o ū u at the time of the Roman conquest. By around the fourth century however, both had transformed this into a vowel system along the lines of *
i ɪ e ɛ a æ o ɔ y ʊ, with length now being conditioned by stress and syllable structure rather than being contrastive and inherent to the vowel.
From the point of morphosyntax, there are three main points where Latin influence on Brythonic is clear. Further, it's clear that the kind of Latin which influenced Brythonic was not the Latin of scholarship or the church, but the spoken everyday Latin of the people. The implication here, of course, is not Catholic missionaries intoning sermons and prayers in good ecclesiastical Latin from which the uncomprehending peasants picked up a word or two, but rather of long-term bilingualism with strong evidence of language shift in favour of Latin.
Firstly, Brythonic, along with early Romance, lost its case system (again, note that Irish, where the Latin influence was solely one of Catholic missionaries intoning sermons, retains a case system to this day). Furthermore, the manner in which the case system was lost seems to have closely paralleled that of early Romance: from a six-case system we see a collapse into a two-case system.
Secondly, like Romance, Brythonic lost the neuter gender. Again, Old Irish maintained a three-gender system until the beginning of the Middle Irish period. The manner in which this happened is probably similar to how it happened in Romance again: via the loss of nasal vowels in polysyllables.
Thirdly, and significantly, Brythonic developed a synthetic pluperfect tense (a synthetic tense is one where you change the end of a verb, not one where you add an auxiliary verb: in French, for example,
je chantais 'I was singing' is synthetic while
j'ai chanté 'I sang' is analytic). This has no parallel in the other Celtic languages and appears to be relatively late. Creating a whole new synthetic tense is unusual enough but the really significant thing here is that the new Brythonic pluperfect was formed
in exactly the same way as the Latin equivalent: by adding the imperfect forms of the verb to be to the perfect stem. For example, in Latin we have the verb
aget 'he acts', which has the perfect stem
ēg-, as in
ēgit 'he has acted'. To form the pluperfect, we add the imperfect form of the verb to be- in this case
erat 'he was' to the perfect stem:
ēgeram 'I had acted'. In early Brythonic then we have the verb *
aget 'he drives', which has the perfect stem *
axt-, as in *
axte 'he has driven'. So we add the imperfect of the verb to be *
ējat 'he was' to the perfect stem giving *
axtējat. Or, in Modern Literary Welsh:
â 'he goes',
aeth 'he has gone' and
aethai 'he had gone'.
tl;dr - yeah, there's actually loads of evidence.