Well, you're not wrong about the common root; there is definitely a root linking Hindi and the Germanic languages (English included), since they're all Indo-European languages, but you have to go back much farther than the Anglo-Saxon invasions of Britain to find that root. It would be found somewhere around six thousand years ago when the original Indo-Europeans started spreading out from somewhere in western Asia (Anatolia, or maybe the Caspian steppes; there are conflicting theories) and expanding west into Europe as well as further east into Asia. The ones who went east into Asia would eventually give rise to the Indo-Iranian branch, of which Hindi is a member, while the ones who headed west into Europe would eventually give rise to the Germanic branch (which English belongs to) and several other branches.
As for the pre-Anglo-Saxon inhabitants of the British Isles, they were Celts, so their native languages (the Brythonic languages) belonged to the Celtic group, which is another branch of the Indo-European family. But these Celtic languages are not the forefather of what eventually became English; English actually developed from the Germanic languages that the Anglo-Saxons brought with when they invaded, rather than from the Celtic languages that were spoken in the British Isles before they arrived. The words "England" and "English" derive from the name of the Angles, who were of course part of the Anglo-Saxon invasion. So the English language is an invention of Germanic invaders, not the "native"* peoples who lived there before; the pre-Anglo-Saxon languages of the British Isles have nothing to do with English.
*I say "native" because the British Celts themselves were not truly native to the British Isles either, but had come over from the continent much earlier, invading/breeding with the pre-Celtic peoples living there. Even those pre-Celtic peoples had migrated across at some point in the past... You see how it goes on.
Howdy-do.
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