Perhaps almost everyone: " Some experts estimate that 80 percent of England's present population descends from Edward III." [1] There was an Economist article posted to HN several months back that explained the general principle better but I couldn't immediately find it.
One of the things missing from the BBC report there was that this was a mitochondrial DNA test. While many may (on paper) be a descendant of Richard III, illegitimacy is common and matrilineal heritage is the best bet.
