The 19-year-old has spent much of this season on loan at League Two rivals AFC Wimbledon, scoring four goals in nine appearances for the Dons.
"I am delighted to welcome Paul to the club," Scott told the Shots' official website. "I have watched him on many occasions for West Ham's Under-21 team and he impressed. He also did well on loan at AFC Wimbledon earlier this season.
"Paul is keen to do well and wants to score the goals which will help the club preserve its Football League status."
McCallum joined the Hammers in January 2011 after rising through the ranks at non-league Dulwich Hamlet.
The youngster, who recently signed a new two-and-a-half-year deal at Upton Park, is expected to make Aldershot's trip to Bradford on Saturday.