Oswell, 24, who scored 22 goals in 2016-17, started his career at Crewe and joined Newtown from Airbus UK in 2014.
"It is disappointing to lose Jason, he has been a fantastic player for us in recent years," Newtown boss Chris Hughes said.
"But he expressed desire to return to the English system."
Welsh Premier League top scorer Jason Oswell has left Newtown for National League North side Stockport County.