No, Cheddar Man's culture would not have been ancestral to any modern English peoples.
First off, while its kind of fun to see, the man's body coloring isn't really any more significant than the color of his clothes. His people likely had issues with Rickets (as did early modern Londoners due to the smog), but that's about the only impact it would have had on his life.
His culture was not directly ancestral to any extant English group. Celtic culture (Welsh, Cornish, etc.) probably did not make it to England until around 800 BC. I've seen some talk about there perhaps being a relationship to the more enigmatic Picts, but the current thinking is that their language was most likely Celtic as well.
The Anglo-Saxons were even later, of course. In fact, 9,000 years ago, not only did Celtic not exist yet, but its theoretical parent language of proto-Indo-European probably was a couple of millennia in the future. When it did come into being, its speakers are thought to have lived in western Asia.
There are probably some traces of his culture that had a modern impact though. The most intriguing example is a possible connection with Stonehenge. Well, at least the site of Stonehenge. Some post-holes have been found nearby that were dated to about 8,000 BC. Not only is that right around the time Cheddar was alive, but Stonehenge is less than 50 miles from Cheddar. Nobody is sure if there was any ritual significance to their placement, but whatever it was, Cheddar Man's culture probably did it.