The 21-year-old, who progressed through the youth systems of West Bromwich Albion and Aston Villa, has made the temporary switch from St Andrew's.
Thorndike is eligible to make his United debut against Bradford Park Avenue on Saturday – and will wear the number 8 shirt.
Notts County loanee Luther Munakandafa, meanwhile, has been recalled by the Meadow Lane club - and will spend the remainder of the season on loan at Northern Premier League promotion chasers Gainsborough Trinity.
The 18-year-old made seven substitute appearances for the Pilgrims - and departs with the club's best wishes for the future.