McGilp, 20, who was born in Glasgow but grew up in Australia, came to Blues on a short-term professional contract in August 2018 after leaving A-League side Melbourne Victory.
Earlier this month he penned an extension which was due to until the end of the 2018/19 campaign.
McGilp has been a regular for Richard Beale's Under-23s since joining Blues and racked up 22 appearances prior to leaving the Club.
Swindon, who are managed by Richie Wellens, are currently placed 12th in League One, seven points off a play-off spot.
McGilp joined Blues as an Under-23s player, in compliance with EFL restrictions, and the Club have agreed a sell-on percentage should he move again for a fee.
He has signed a two-and-a-half year contract with Swindon.