The 18-year-old defender is set to be on the bench for Friday night's game against Rochdale.
He is highly-rated by the Premier League leaders and captained the Academy side throughout the 2011/12 season.
"We are delighted to have brought a young talent in from Manchester United," Mark Yates told the club's official website.
"Luke is a proper defender and dominant. Neil Howarth (assistant manager) has done well in getting the deal to go forward and I would like to thank Manchester United for their help.
"It will be a big test for Luke but one I'm sure he will cope well with."
McCullough has represented his country at Under-16 and Under-17 level and has been likened to Jonny Evans.