Waite has shone in two seasons with the Robins, scoring 14 goals for their academy team last season and six for first team this campaign as they challenge for the title in the Evo-Stik First Division South.
The 17-year-old has moved up five divisions to joins Notts in League One and could even figure in the Magpies' last eight games of the season as they chase promotion to the Championship.
"I'm really happy to be at Notts because it's a big club, playing in a good league. It's time for me to step up a level and improve my game," Waite said.
Magpies' chief executive Jim Rodwell said the club will continue to try and sign promising young players.
"We're delighted that Tyrell has signed," he said. "It really is about buying into potential and one of the remits the manager was given is that we intend to make it a priority to sign young, emerging talent."