I had heard that, too. And analogous examples are given of Irish English where the syntax is affected by that of the Irish language. And it would make perfect sense especially with some of the migration theories that suggest that the number of Germanic people coming over was relatively low and they immediately occupied the top social rungs (much as the Normans) and the subject people or peoples completely adopted the Germanic culture and language .
I don't know much about Old Welsh, but as I have said, I have been long been struck by the similarity in word order and tense formation between Old Norse (and indeed modern Swedish, Norwegian and Danish) and Modern English. My difficulty in reading Old Norse is down to the fact I haven't put in nearly as much time learning vocab as I have with Old English, but the sentence structure is fairly straightforward due to it's similarity to modern English, whereas my difficulty reading Old English is often down to the unfamiliar construction of the sentences.
Also, I remember reading somewhere that Modern English has particle verbs composed of a verb and 2 particles - get along with, put up with, look forward to - apparently English is the only West Germanic language that does this, whereas all the Norse languages do this. The last example has a counterpart in Swedish - ser fram emot. Also we have separable particle verbs eg "Look the word up in the dictionary." Norse languages do this. West Germanic languages, all apart from this insular renegade, like to keep their phrasal verbs tightly bound together, so much so that they often don't write them as separate words.
I'm not discounting the idea of a significant British influence on English, just that there were a number of languages present and contributing to this monstrous chimera we now speak.
A couple of years ago, I was thinking about the word bloke. That's clearly not Germanic and I was wondering if it was British in origin. I checked out the word in the Collin's dictionary, which has an excellent etymology, and it turns out it's from a language called Shelta, a language spoken by Irish travelers: a Goidelic language.