Desmond walked out of the tube station, past Charlie, paying him no notice. When it started to rain, Charlie packed up his guitar and ran off. Then he saw the mugging, ran into the alley and chased off the purse snatcher.
As things played out in the version of the events as Desmond relived them after the portal exploded, Desmond saw Charlie, confronted him and delayed him long enough so that he likely never witnessed the mugging.
Now events are falling into place as Desmond predicts them, except that Charlie doesn’t die. Desmond’s flashes of Charlie’s deaths follow his relived version of the events. But due to some twist of causality in the way that things actually played out (or simply due to karma Charlie earned by preventing the mugging), Charlie doesn’t die.