Right-back Byrne, 21, has signed for the Robins on a free transfer after impressing in the seven appearances he made during a temporary spell at the County Ground last season.
Midfielder Pritchard, 20, is a highly-rated prospect at White Hart Lane and was recently part of the England Under-20 World Cup squad.
A Swindon statement on Twitter: read: "Swindon Town have today signed Tottenham Hotspur's Alex Pritchard and Nathan Byrne #STFC.
"Former loanee Byrne has signed a two-year deal while Pritchard joins Town on a season-long loan #STFC."
Swindon finished sixth in League One last season but missed out on promotion following defeat to Brentford in the playoff semi-finals.