The claim behind this is fairly narrowly specific. Namely the distinction in pronunciation between "pass", "bath", "laugh" is sometimes attributed to French influence, since French also saw a lengthening of some sounds before at least -s. The s has since been removed, and the length distinction is weak in metropolitan France. Some example words and earlier English borrowings are castle — château, paste — pâte.

It is unlikely to be correct. The environments, though related, are non-identical. The vowel systems were radically different - English had some long vowels and some short vowels. French had only one length. English actually had a bit of a whole in its vowel system at that location - long vowel and short vowels everywhere except for the area round "trap", "Goth", when there were two short vowels and, recently, no long vowels (since "name" had moved forwards and kept moving forwards till it was caught into the vortex of merging sounds that created the modern pronunciation of words like day, they, weigh, break - once all pronounced differently).

At the exact same time, English was witnessing various other changes. The vowel in "Goth" was low at the time, and lengthened too. The vowel in "water", "what" (lengthend and) retracted. The vowel in "bar" failed to change when the vowel in "bat" moved forwards. Some of the changes were motivated at filling the hole introduced by the merger of "name" with "rain".

Indeed, this means it actually happened _prior_ to the American split, since these changes are present in America too. And it is, partially, represented in American English ("father" if taken narrowly, or the fact that many Americans who distinguish "cot" and "caught" say "caugh", "long", "dog" as "cawff", "lawng", "dawg" — and indeed, if you believe one credible hypothesis, the fact that so many Americans have merged "cot" and "caught" because there was too much variation between those sounds).

In both America and English, there was options. America ended up standardising on lengthened Goth, shortened path. England, to the extent that it standardised, did so on shortened Goth, lengthened path. But these happened at around the same time from roots planted prior to the linguistic separation of the two countries.

So you can see it seems the English change was wholly internally motivated. The only possible exception is that another major source of the broad a - in words like "dance" and and "enchanted" - was indeed an attempt to mimic the French nasal vowels. Ever since these words were adopted in the middle ages, some people had said them as "daunce" and "enchaunted" and "laundry" and "haunted" and "aunt" and some people had said them as a "dance" and "enchanted" and "landry" and "hanted" and "ant". In those days, "au" was pronunced like in German - maybe how you say "how". And you can see that the spellings don't have any particular preference in coming from one pronunciation scheme or the other. The standard English accent derives from the pronunciations that did not always feed the spelling, and the standard American accent derives from pronunciations that di not always feed the spelling. The only reason it's slightly obscure is that the sound spelt "au" sometimes became "ah" and sometimes became the modern English "aw" sound. In "calm", it became "ah". In "walk", it became "aw". So the standard British choice was "daunce", "enchaunted", spelt as "dance", "enchanted", and pronounced with the variant of "calm". The only bit of copying the French came centuries before the American colonies.

The true history of the Early Modern English low vowels remains to be written. But that's as fair a summary of it as you'll find on the social medias.

The Americans deliberately copied the French meaning of "billion", and then the French, seeing the Americans were using it, threw it away and stole the British meaning of "billion". The British held onto their tradition for a while, but they have no strong desire to align with the French, so it was really only a matter of time till, in the last generations, they adopted the American meaning.