Picking up an item is neither faster nor easier because a creature's prone
A prone creature doesn't pick up an item in less time than a standing creature, and a prone creature still provokes an attack of opportunity from each foe that threatens the creature when it attempts to pick up an item.
Despite the weapon being right there on the ground and in the prone creature's space, if the creature has been the victim of the combat maneuver disarm, then the typical creature must take a move action that provokes attacks of opportunity to pick up the weapon. (Also see the Table: Actions in Combat and its short accompanying FAQ entry.)
Note: The fighter in question has picked here the right order in which to execute these maneuvers: If the fighter's trip combat maneuver succeeds and renders the foe prone, when the fighter makes the disarm attempt against the prone foe, Combat Modifiers has the foe suffer a −4 penalty to his Combat Maneuver Defense score as "[a]ny penalties to a creature's AC also apply to its CMD.")