In technical terms, it's because there was no local Romance language in Britain at the time these peoples invaded Britain. The two languages in most use at the time were Brythonic (the Celtic language that eventually became Welsh and Cornish) and Latin, the latter of which was not a spoken language outside of religious society.
Why Brythonic was not adopted to any significant degree into the languages that would become Old English is one I can't answer, and I'd love to hear from one of our Anglo-Saxonists on that. The lack of word-borrowing that happened between the Germanic and Celtic peoples is part of why we know that the Anglo-Saxon settlement of Britain was not a conquest, but a displacement of the native peoples. The settlers pushed the Brythonic peoples out of their way with apparently not much interest in absorbing them or integrating themselves with them in any way that we have evidence of. I want to make that distinction clear because it was a very different case with the Scandinavian settlers who colonized northern England in the 9th century. Scandinavian men took Anglo-Saxon wives and to some degree worked with Anglo-Saxon laborers, and as a result, there are a lot of English loan-words with Scandinavian origins (household words like fork or kettle and natural words like dirt).
Even more to your question, though: a Classicist can definitely answer better than I can here, but the Latin that the Franks, Visigoths, and Vandals adopted was the same Latin the Angles, Saxons, and Jutes adopted before invading Britain. Latin did heavily influence Old English in several waves, both during the Roman period before the invasion and in the context of clerical reforms later. It sounds like what you're asking is "if these are all settlers who adopted Latin, why does English sound so different," and if so, one big factor is the simple fact that the Anglo-Saxon invaders settled an island and their language developed relatively isolated from continental Europe. We don't see the same kind of permeability with the Celtic languages besides a few place-names, so besides the Latin-influenced West Germanic languages the settlers crossed the channel with, the only major linguistic influence Old English faced was that of the Scandinavian invaders.
Of course, then 1066 and William the Conqueror happened, which is where the development of English really gets interesting, but that's way past the scope of your question. But my understanding of why the Anglo-Saxons didn't absorb Brythonic is too far outside what I know to answer confidently, so Classicists, I'd love to hear from you.
EDIT: Looks like I knew less about this than I thought! Many thanks to u/Steelcan909 and u/BRIStoneman for their corrections. Tagging u/LigniteCA to make sure they see the replies.