The 20-year-old spent two spells out on loan last season with Aldershot and then Exeter where he made 26 starts in League One.
Taylor, who is a regular for his country's U21s, will stay at Whaddon Road until January 2.
The Robins have also extended the loan of Taylor's Royals team-mate Lawson D'Ath until the same date. The midfielder joined last month but played just twice before picking up an injury.
The 19-year-old will miss Cheltenham's home match with strugglers Barnet but should return for their FA Cup clash with Hereford.