What language would I speak if the Anglo-Saxon's hadn't arrived?

Or may be called Old Welsh (hen Cymraeg). We have no real way of knowing at this time whether the language spoken in Cumbria came to be unintelligible to speakers in what is now Wales before it finally became extinct. When the Kingdom of Strathclyde fell, it is said there was an exodus of its people to Gwynnedd, where presumably they felt they would find fellow-speakers of Welsh.

Continental Celtic was quite different from the Insular Celtic branches – it didn’t have a lot of the innovations the Insular varieties had, and was much closer to Latin, clearly showing its kinship to the Italic languages. See here, for example, or here for examples of Gaulish texts. The fact that Gaulish and Latin were so similar meant it would have been easy for Gauls to learn Latin, and as Latin was a much more useful language for commerce, politics, etc., it may have hastened the decline of Gaulish as people chose to use Latin instead.

Dog is derrived from *dogge *(M.E.) which is derived from *docga *(O.E.). I don’t think much is known about where the O.E. word came from, and I’m unsure about how far back into O.E. we can trace docga.

There’d be another option, I think; when a trade language becomes a common second language, or is used by people with varying dialects to communicate across local borders. It’ll sometimes be hard to separate this out from the effects of colonization; but in the hypothetical case of the OP, one would have to ask if, at some point, the isolated Celts of Britain found themselves learning something else so they could buy digital watches. (Or whatever.)

The word dog likely goes back to the beginnings of Germanic language, if not earlier. A cognate exists in German, Dogge, but in that language refers to a particular kind of dog, a mastiff. (In the same way, the German word for dog generally, Hund is a cognate of the English word Hound, which itselfs denotes only a certain category of dog.

IME the fact that the meanings have diverged is strong evidence that the words evolved separately in each language. If they were more recent borrowings, wouldn’t the meanings be closer?