The nature of the loan also has a 28-day recall in the agreement.
The midfielder signed his first profession deal in the summer and was one of the goalscorers at Old Trafford in the FA Youth Cup last season.
Robertson was also nominated for the April 2022 League Football Education goal of the month for his strike in the 5-2 home win over Huddersfield Town.
It was his side's fourth goal of that game, as a wonderful team move culminated in Nat Wallace picking out the unmarked Robertson at the back post, who smashed home his third goal of the season.
He joins the side officially in time to host Ilkeston Town in their season opener on Saturday after their match away at Hebburn Town last weekend was called off due to a waterlogged pitch, while he has played three games as a trialist in pre-season against Darlington, Blyth Spartans and Scarborough Athletic.