The 18-year-old midfielder, who never made a senior appearance while at the Britannia Stadium, will join the Cobblers on 1 July.
Boss Ian Sampson told BBC Radio Northampton: "He's got a dominance about him. For someone of such a young age he's got a physical frame on him.
"He's left-footed which gives us good balance and he's an all-round athlete."
Wedderburn had three months on loan at Notts County in the 2008/09 season, making 11 appearances, before another loan switch in November to Hereford, where he featured three times.
He played for Hereford against Sampson's men in League Two in December last season, with the Bulls coming out 3-1 winners.
The midfielder fits into Sampson's blueprint of taking on young players he can help develop, but the Cobblers manager is wary a successful season may see the youngster attracting the attention of other clubs.
"Hopefully we will get promoted and his stock will rise and he may move on, but hopefully he'll stay with us," he said.
"Our objective is to get promoted but to do it in a slightly different way and it means bringing younger players in and trying to develop them."
Wedderburn, who can also play at centre-half, has three England Under-19s caps and was a regular for Stoke's youth and reserve sides.
Now he says he is ready to take on the challenge of League Two football.
He said: "It's a physical battle; it's difficult to get the ball down and play.
"After being released I just wanted to find another club and I know this is a club that's pushing forwards.
"I just want to play as many games as I can."
Meanwhile, Sampson has confirmed that defender Abdul Osman and midfielder Liam Davis have yet to accept the club's offer of new contracts.
Source: BBC Sport
Source: BBC Sport