The England Under-19 international has made seven appearances for Albion since coming through their academy and spent two spells on loan with Portsmouth, one of those in the Championship last season.
He became West Brom's youngest player ever to represent the club when he replaced Simon Cox as a substitute aged 16 years, 328 days to make his senior debut against Sheffield Wednesday.
Peterborough boss Darren Ferguson said: "He is a very good player, he went out on loan to Portsmouth last season and did well and he is a player that I have seen quite a few times now.
"He is a good size, a very good footballer and it gives us more strength in midfield.
"I know this is an area that we have got numbers in, but I think this offers something slightly different. It will allow us to put Michael Bostwick in at centre half, which is something that I have been thinking about for a while.
"It frees up more options for me because I think we need another centre half and this allows me to utilise 'Bozzy' there. That was a big attraction to me.
"George is highly thought of at West Brom and we are pleased to get him because a few other clubs were in for his services, but I know he has had conversations with Saido Berahino and he said how much he is enjoying his time with us and that tipped it for us. I am looking forward to working with him."